## Semi-analytical method for light interaction with 1D-periodic nanoplasmonic structures

Optics Express, Vol. 16, Issue 12, pp. 8938-8957 (2008)

http://dx.doi.org/10.1364/OE.16.008938

Acrobat PDF (2833 KB)

### Abstract

We present a detailed description of a computationally efficient, semi-analytical method (SAM) to calculate the electomagnetic field distribution in a 1D-periodic, subwavelength-structured metal film placed between dielectric substrates. The method is roughly three orders of magnitude faster than the finite-element method (FEM). SAM is used to study the resonant transmission of light through nanoplasmonic structures, and to analyze the role of fundamental and higher-order Bloch surface plasmons in transmission enhancement. The method is also suitable for solving the eigenvalue problem and finding modes of the structure. Results obtained with SAM, FEM, and the finite-difference time-domain method show very good agreement for various parameters of the structure.

© 2008 Optical Society of America

## 1. Introduction

3. J. Homola, ed., *Surface Plasmon Resonance Based Sensors* (Springer, 2006). [CrossRef]

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

7. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. **83**, 2845–2848 (1999). [CrossRef]

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

15. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101–202 (2007). [CrossRef]

17. J. L. Volakis, A. Chatterjee, and J. L. Kempel, *Finite Element Method for Electromagnetics* (IEEE Press, New York, 1998). [CrossRef]

18. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Prop. **14**, 302–307 (1966). [CrossRef]

10. A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B **67**, 195402 (2003). [CrossRef]

19. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “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]

9. S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B **67**, 035424 (2003). [CrossRef]

24. S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B **70**, 075103 (2004). [CrossRef]

*N*in the field expansion. The study of HO BSPs also calls for a more complete field expansion. An extension of the method [24

24. S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B **70**, 075103 (2004). [CrossRef]

*N*>1 harmonics has been conceptually outlined in [25

25. A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express **13**, 8730–8745 (2005). [CrossRef] [PubMed]

*N*=1, raising concerns over applicability of the method to true lamellar gratings [26

26. E. Popov and M. Nevière, “Analytical model of the optical response of periodically structured metallic films: Comment,” Opt. Express **14**, 6583–6587 (2006). [CrossRef] [PubMed]

24. S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B **70**, 075103 (2004). [CrossRef]

## 2. 1D-periodic nanoplasmonic structure

*ρ*, thickness

*h*, and the contrast Δ

*ε*=ℜ(

*ε*

_{T}-

*ε*

_{B})>0. In this work we mostly focus on planar bimetallic gratings with ℜ(

*ε*

_{T})<0. However, we also present some encouraging results when applying the SAM to a true slit geometry with

*ℜ*

_{T}=1. Unlike metal-dielectric (true slit) structures, where both evanescent and waveguiding modes contribute to the transmitted light [7

7. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. **83**, 2845–2848 (1999). [CrossRef]

13. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express **13**, 4485–4491 (2005). [CrossRef] [PubMed]

14. N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. **9**, 490–495 (2007). [CrossRef]

*ε*) appear in the transmitted spectrum as compared to the spectra of slit arrays. This feature is essential for nonlinear optical and sensing applications. The described modulation of the metal’s permittivity can be achieved, e.g., by alternating stripes of two metals that are optically different in some wavelength range, such as gold and silver in the visible. For other possible combinations, see Fig. 2. Other options to create bimetallic gratings may include geometric structuring to modify the effective permittivity of a stripe and using alloys or metal-dielectric composites having a prescribed effective permittivity [27

27. W. Cai, D. A. Genov, and V.M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B **72**, 193101 (2005). [CrossRef]

28. M. I. Markovic and A. D. Rakic, “Determination of the reflection coefficients of laser light of wavelengths λ (0.22 *µ*m, 200 *µ*m) from the surface of aluminium using the Lorentz-Drude model,” Appl. Opt. **29**, 3479–3483 (1990). [CrossRef] [PubMed]

_{p}=145 nm and

*ε*

^{∞}

_{B}=1.53 correspond to gold [29

29. P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. **125**, 164705 (2006). [CrossRef] [PubMed]

_{p}varies between 10

^{7}nm and 5×10

^{5}nm, where the latter value corresponds to ℑ(

*ε*

_{T,B})≈0.1 at 900 nm. We also show several results for

*γ*

_{p}=4×10

^{4}nm, which gives realistic loss in metal.

## 3. The semi-analytical method: details

### 3.1. Maxwell’s equations for the TM-mode and the periodic ansatz

*ε*

_{0}is the average permittivity of the film (not to be confused with the free-space permittivity) and

*g*=2

*π*/Λ is the magnitude of the wave vector of the grating. Without loss of generality, we only consider the even harmonics, since the origin of

*x*-axis can be chosen arbitrarily (Fig. 1). For the stepwise function (1), the Fourier coefficients are

*ε*

_{0}) as the average loss of the film and set ℑ(

*ε*)=0. A TE mode does not participate in the resonant interaction with the structure [24

_{m}**70**, 075103 (2004). [CrossRef]

**E**=(

*E*

_{x},0,

*E*)

_{z}*e*+c.c. and

^{-iwt}**H**=(0,

*H*

_{y},0)

*e*+c.c. After substitution in Maxwell’s equations

_{-iwt}*E*

_{x}and

*E*

_{z}

*Z*=376.6 Ohms is the free space impedance.

*z*=0) and the bottom (

*z*=

*-h*) interfaces, respectively. The unknown coefficients

*A*

_{σ},

*P*

_{σn},

*Q*

_{σn},

*a*

_{σ},

*p*

_{σn},

*q*

_{σn}and values of the wave parameter

*k*are to be found from (5) and from the electromagnetic boundary conditions. For consistency, the number of harmonics retained in the ansatz (7) should be no less than the number of harmonics in the representation of the modulated medium (3), i.e.

*N*≥

*M*. Unless otherwise stated, we assume

*M*=

*N*. For the superstrate (substrate), only the first (second) term in (7) represents the general solution.

### 3.2. Solution in the structured metal film, -h<z<0

*ngx*), sin(

*ngx*), and the free terms, we arrive at two decoupled systems of linear algebraic equations. In a matrix form, these systems can be written as

**U**

_{N}and

**V**

_{N}are square matrices of size 2

*N*+1,

**=(**

*ξ**A*

_{x},

*P*

_{x1},

*Q*

_{z1},

*P*

_{x2},

*Q*

_{z2}, …,

*P*,

_{xN}*Q*)

_{zN}^{T},

**ζ**=(

*A*

_{z},

*P*

_{z1},

*Q*

_{x1},

*P*

_{z2},

*Q*

_{x2}, …,

*P*

_{zN},

*Q*

_{xN})

^{T}. The analysis is simplified for the normal incidence of light because the incoming plane wave lacks any

*z*-component of the field. Therefore, none of the components

*A*

_{z},

*P*

_{zn}, and

*Q*

_{xn}are excited within the structure, resulting in

**U**

_{N}is a sum of two matrices

**Y**

_{N}and

**u**

_{N}:

*ε*(

*x*) with the field, and from the terms having the second-order mixed derivatives in Maxwell’s Eq. (5).

**Y**

_{N}are

*n*,

*m*=1,2…,

*N*and

*f*

^{A}

_{m}and

*f*

^{B}

_{m}, respectively; Θ(

*x*)=1 for

*x*≥0 and Θ(

*x*)=0 otherwise. The other matrix in (11),

**u**

_{N}, has the following nonzero elements

*N*=2 we obtain

*k*along the

*z*-axis, resulting in an (

*N*+1)th order polynomial for

*k*

^{2}, whose zeros are ±

*k*,

_{j}*j*=1, …,

*N*+1. For a more efficient numerical treatment, one can recast (15) in the form

**A**and

**B**are

*a*

_{11}=

*a*

_{2n,2n}=1 and

*b*

_{2n+1,2n}=-

*b*

_{2n,2n+1}=

*g*, respectively. Matrix

**C**equals

*K*

^{2}

**Y**

_{N}with some extra terms added on the main diagonal, namely

*c*

_{2n+1,2n+1}=

*K*

^{2}

*y*

_{2n+1,2n+1}-

*n*

^{2}

*g*

^{2}. An elaborate discussion about the methods for solving (16) can be found in [31

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

^{®}[32].

*k*=±

*κ*of

_{j}**U**

_{N}corresponds to an independent solution of (7). A linear superposition of all solutions gives the general form of the electric field distribution in the film. For a bounded medium (-

*h*<

*z*<0), any sign of the eigenvalue results in an acceptable solution. We keep the “+” sign for ℜ(

*κ*)>0 and, accounting for (10), obtain from (7)

_{j}*n*shows the order of the harmonic. The amplitudes

*A*

^{(j)}

_{x},

*a*

^{(j)}

_{x},

*P*

^{(j)}

_{xn},

*p*

^{(j)}

_{xn},

*Q*

^{(j)}

_{zn},

*q*

^{(j)}

_{zn}are to be determined by the boundary conditions.

### 3.3. Solution in the superstrate, z>0

*z*>0. Matrix

**U**

_{N}is considerably simplified in the absence of periodic variation of permittivity in the dielectric medium. With

*ε*

_{0}replaced by

*ε*

^{+}and any other

*ε*≡ 0, it becomes block-structured, i.e. it is reduced to

_{m}*N*+1 independent equations for each harmonic. The corresponding eigenvalues are given by

*n*=0,1, …,

*N*. The zero-order eigenvalue -

*ε*

^{+}

*K*

^{2}corresponds to plane wave solutions for the

*E*

^{+}

_{x}component of the field, namely

*R*

_{x},

*P*

^{+}

_{xn}, and

*Q*

^{+}

_{zn}being the unknown amplitudes to be determined by the boundary conditions.

*s*±=±1 is the sign of the eigenvalue. It must be chosen properly to obtain physically meaningful solutions. For a lossless dielectric,

*n*

^{2}

*g*

^{2}-

*ε*

^{+}

*K*

^{2}can be either positive (evanescent harmonic) or negative (propagating harmonic). In the superstrate, the amplitude of the evanescent wave should decay for

*z*→+∞, which imposes

*s*+=-1. To ensure the correct sign of the phase for reflected harmonics (i.e. make the phase front propagate in the direction of the wave vector), one has to take

*s*+=1 if

*n*

^{2}

*g*

^{2}-

*ε*

^{+}

*K*

^{2}<0. Therefore, for the purely real

*ε*

^{+}

*s*

^{±}occurs due to discontinuity of the square root function across its branch-cut along the negative real axis. If the loss in the dielectric is taken into account, i.e. ℑ(

*ε*

^{+})>0, then (

*η*

^{+}

_{n})

^{2}is complex-valued. For this case,

*s*

^{+}= -1 for both propagating and evanescent waves will result in the correct behavior of both evanescent and propagating harmonics (Fig. 3, left plot).

*k*=

*η*in matrix

_{n}**U**

_{N}for the superstrate, with a subsequent solution of (15), gives a relation between the harmonic amplitudes

### 3.4. Solution in the substrate, z<-h

*T*is the amplitude of the zero-order diffracted wave. For a lossless dielectric, i.e. with ℑ(

_{x}*ε*

^{-})=0 we have

### 3.5. Boundary conditions

*κ*(for the film) and

_{j}*η*

^{±}

_{n}(for dielectric media). Next, we apply boundary conditions to uniquely determine the unknown amplitudes. The electromagnetic boundary conditions at

*z*=0 are

9. S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B **67**, 035424 (2003). [CrossRef]

*z*=-

*h*:

### 3.6. The main matrix equation

**U**

_{N}as follows. Because of (15), the equations of (8) are linearly dependent. Therefore, one equation in (8), e.g., first, can be dropped. Then, the remaining system can be solved to find the harmonic amplitudes

*P*

^{(j)}

_{xn}and

*Q*

^{(j)}

_{zn}expressed through the zero-order amplitude

*A*

^{(j)}

_{x}. In other words, we define

*α*and

_{nj}*β*as

_{nj}**r**=-2

*K*

^{2}(

*ε*

_{1}, 0,

*ε*

_{2}, 0, …,

*ε*, 0)

_{N}^{T}is obtained from the first column of matrix

**U**

_{N},

**=(**

*ρ′**α*

_{1j},

*β*

_{1j},

*α*

_{2j},

*β*

_{2j}, …,

*α*,

_{Nj}*β*)

_{Nj}^{T}and

**W**

^{(j)}

_{N}(

*κ*) is obtained from

_{j}**U**

_{N}by removing the first row and the first column and substituting

*k*=

*k*. For example, for

_{j}*N*=2

*κ*in (34) with -

_{j}*κ*to solve the system for

_{j}**ρ″**=(

α ˜

_{1j},

β ˜

_{1j},

α ˜

_{2j},

β ˜ _{2j}, …,

*,*α ˜

_{Nj}*)*β ˜

_{Nj}^{T}where

**and**

*ρ′***are known, we can express all unknown amplitudes in (17) through 2**

*ρ″**N*+2 zero-order harmonics that form a vector

*N*equations (for each

*n*), where unknown are the elements of vector

**a**. A similar relation can be obtained from (30b), (31), and (25), which gives

*N*additional equations for the total of 2

*N*+2 elements of vector

**a**. To complete the set, the remaining two equations are obtained by substituting

*T*

_{x}from (30a) in (32) to obtain

*N*+2 equations for the elements of vector

**a**can be written in matrix form as

**v**is

**M**is

**m**is

*N*×

*N*+1, with

*p*and

*q*being the row and the column indices, respectively,

*p*=1, …,

*N*,

*q*=1, …,

*N*+1. The submatrices have the following elements

*A*

^{(j)}

_{x}and of

*a*

^{(j)}

_{x}, respectively. The last two rows of (41) originate from (38) and (39), respectively. Numerical solution of (40) essentially completes the computational algorithm of the SAM.

### 3.7. Electromagnetic field distribution

**a**are known, one can easily calculate the electromagnetic field for the whole structure. As follows from (17), (33), and (35), the field components in the metal film (-

*h*≤

*z*≤0) expressed through the elements of vector

**a**are

*z*>0) are then

### 3.8. Transmission and reflection coefficients

*T*

_{x}and

*R*

_{x}contribute to the total transmitted and reflected power. Because

*η*

^{±}

_{n}in (19) and (23) are purely imaginary for all

*n*, all higher-order harmonics are evanescent and do not contribute to the total field at

*z*=±∞. In this case, the zeroth-order transmitted amplitude is given by (30a), while the zeroth-order reflected amplitude is obtained from (27a) as

*z*-component of the Poynting vector integrated over the period ∫

_{Λ}

**S**

_{z}d

*x*|

_{z=-∞}to the same quantity that is calculated for the incident wave at

*z*=+∞.

*z*-component of the Poynting vector

*ε*-≥(

*n*λ/Λ)

^{2}is satisfied for

*n*=1,2,…,

*L*, i.e.

*L*≥1 harmonics in the substrate are propagating. In obtaining (49), we have used that ∫

_{Λ}Σ

_{n}cos(

*ngx*)Σ

_{m}cos(

*mgx*)d

*x*equals Λ/2 if

*m*=

*n*and zero if

*m*≠

*n*. By using (25) and (22), formula (49) can be simplified to

### 3.9. Computation steps of the SAM: a summary

- obtain
*ε*_{0}and*N*harmonic coefficients that represent the grating (3), - solve the main matrix equation (40) to find vector
**a**,

## 4. Results and discussion

^{®}[33] was used as FEM implementation. It has been run on a 2-processor 4 GB workstation. FDTD calculations were performed on an 8-processor, 32 GB workstation while the SAM code runs on a standard laptop with 1 GB memory and 1.8 GHz processor. This section also features fundamental and higher-order Bloch surface plasmons and the SAM modeling of a true slit grating.

### 4.1. Validation of the SAM with FEM and FDTD

34. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

*h*=72 nm the transmission drops to nearly -60 dB (i.e. 10

^{-6}), which is four orders of magnitude below the tunneling rate. Such regimes require a high-resolution computation grid and are quite difficult to resolve, especially with the FDTD method. As a result, the FDTD solutions of Figs. 4(b),(c) are not fully convergent to the solution. Nevertheless, good agreement between the SAM and the FEM is seen in Figs. 4(b)(c), even for very low transmittance levels.

*N*one needs to keep in the field expansion depends on the system parameters. We have found that for low contrast bimetallic gratings,

*N*can be quite low. In fact, for Δ

*ε*=5 just one harmonic is sufficient (Fig. 5). This leads us to a non-intuitive conclusion that the response of a true rectangular but

*low-contrast*bimetallic grating with

*ε*

_{T,B}<0 can be accurately modeled by response of a sinusoidal (phase) grating. On the contrary, even the phase grating itself (

*M*≡1) might require

*N*>1 terms in the filed expansion if the amplitude of modulation

*ε*

_{1}is high (Fig. 6).

### 4.2. Fundamental Bloch surface plasmons

10. A. M. Dykhne, A. K. Sarychev, and V. M. Shalaev, “Resonant transmission through metal films with fabricated and light-induced modulation,” Phys. Rev. B **67**, 195402 (2003). [CrossRef]

9. S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: An analytical study,” Phys. Rev. B **67**, 035424 (2003). [CrossRef]

**70**, 075103 (2004). [CrossRef]

^{res}

_{1}can be estimated from the dispersion relation for the SPP on a smooth metal surface. It is, therefore, given by the solution of the transcendental equation λ=Λ(

*ε*

_{0}(λ)

*ε*

^{±}(λ)/[

*ε*

_{0}(λ)+

*ε*

^{±}(λ)])

^{1/2}. The exact value of λ

^{res}

_{1}is determined by the modal equation det

**M**=0, which for a relatively thin film has two distinct solutions (Fig. 7(a). The solutions are complex-valued even in the absence of any loss in metal [9

**67**, 035424 (2003). [CrossRef]

^{res}

_{1}). This excitation is accompanied by a transmission enhancement of the structure (Fig. 7(b). Note that both values of λ

^{res}

_{1}exactly coincide with the maxima of transmittance. The nonzero imaginary part of λ

^{res}

_{1}reflects the finite lifetime

*τ*=ℜ(λ

^{res})

^{2}/[2

*πc*Im(λ

^{res})] of each BSP mode. Since the lower-frequency mode (λ=918.2 nm) has a shorter lifetime, the right resonance peak on Fig. 7(b) is broader than the left peak. For this simulation, we have taken a close to realistic loss coefficient in the metal with γ

_{p}=4×10

^{4}nm, which corresponds to ℑ(ε

_{T,B})=0.81. Excellent agreement with the FEM can be seen for the calculated transmittance. On the computational side, when loss in a system is high, with a relatively large size of matrix

**M**, the absolute value of det

**M**becomes very large making it difficult to accurately find zeros of the eigenvalue equation.

*E*

_{z}component) can be observed near both resonant wavelengths. Also notable is the behavior of the Poynting vector

**S**in the wavelength region near resonances. There, the the energy flow changes direction along the

*x*-axis and vector

**S**forms circular areas. If the FBSP is not excited, the nonresonant tunneling becomes the only mechanism of light transmission. It is accompanied by a strong reflection from the film, visible as a pronounced interference pattern for the

*E*

_{x}component of the field.

### 4.3. Higher-order Bloch surface plasmons

*x*-axis. The light confinement in the transverse direction is therefore ~λ/16, which is quite promising for the subwavelength lithography. The penetration depth of the localized field is even less (~20 nm). To minimize the spot size for a fixed period of the structure, one has to increase the index of the substrates.

### 4.4. The true slit geometry

*ε*

_{T}=1 in (1). Except for the regime of nearly complete transparency, the quantitative agreement of FEM and FDTD with the SAM is very good. The wavelength of maximum transmittance can be predicted quite accurately with a more accurate ansatz in the SAM. However, the SAM underestimates the magnitude of maximum transmittance 𝒯

_{max}by about 5 dB. The reason for this discrepancy is an incomplete account for loss of metal in the current implementation of the SAM. Taking only the zero-order term in the expansion of the imaginary part of

*ε*(

*x*) is not sufficient to achieve a perfect result. Further development of the method is needed to improve its stability when dealing with the unequal loss of the grating’s components. Highly conductive gratings under TM incidence are known to suffer from numerical instabilities [21

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

20. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A **13**, 779–784 (1996). [CrossRef]

22. 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**, 1019–1023 (1996). [CrossRef]

23. E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A **17**, 1773–1784 (2000). [CrossRef]

36. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A **24**, 3781–3788 (2007). [CrossRef]

_{max}but gives a slightly offset resonant wavelength. The computation time with the SAM, however, remains within several minutes even for metal-dielectric structures with a small duty cycle.

## 5. Conclusions

## Acknowledgments

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12. | D. Gérard, L. Salomon, F.
de Fornel, and A. V. Zayats, “Ridge-enhanced optical transmission through a continuous metal film,” Phys. Rev. B |

13. | Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express |

14. | N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A: Pure Appl. Opt. |

15. | K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. |

16. | A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, “Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates,” J. Opt. Soc. Am. B (submitted). |

17. | J. L. Volakis, A. Chatterjee, and J. L. Kempel, |

18. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Prop. |

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

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

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

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

23. | E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A |

24. | S. A. Darmanyan, M. Nevière, and A. V. Zayats, “Analytical theory of optical transmission through periodically structured metal films via tunnel-coupled surface polariton modes,” Phys. Rev. B |

25. | A. Benabbas, V. Halté, and J.-Y. Bigot, “Analytical model of the optical response of periodically structured metallic films,” Opt. Express |

26. | E. Popov and M. Nevière, “Analytical model of the optical response of periodically structured metallic films: Comment,” Opt. Express |

27. | W. Cai, D. A. Genov, and V.M. Shalaev, “Superlens based metal-dielectric composites,” Phys. Rev. B |

28. | M. I. Markovic and A. D. Rakic, “Determination of the reflection coefficients of laser light of wavelengths λ (0.22 |

29. | P. G. Etchegoin, E. C. L. Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. |

30. | E. Palik and G. Ghosh, eds., |

31. | F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. |

32. | |

33. | |

34. | Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. |

35. | M. Nevière and E. Popov, |

36. | N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A |

**OCIS Codes**

(240.0310) Optics at surfaces : Thin films

(240.6680) Optics at surfaces : Surface plasmons

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: March 7, 2008

Revised Manuscript: May 21, 2008

Manuscript Accepted: May 29, 2008

Published: June 3, 2008

**Citation**

Andrey Kobyakov, Aramais R. Zakharian, Arash Mafi, and Sergey A. Darmanyan, "Semi-analytical method for light
interaction with 1D-periodic
nanoplasmonic structures," Opt. Express **16**, 8938-8957 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8938

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### References

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- N. Garcia and M. Nieto-Vesperinas, "Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits," J. Opt. A: Pure Appl. Opt. 9, 490-495 (2007). [CrossRef]
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- A. Kobyakov, A. Mafi, A. R. Zakharian, and S. A. Darmanyan, "Fundamental and higher-order Bloch surface plasmons in planar bimetallic gratings on silicon and glass substrates," J. Opt. Soc. Am. B (submitted).
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- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell???s equations in isotropic media," IEEE Trans. Antennas and Prop. 14, 302-307 (1966). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "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]
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