OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 5 — Feb. 28, 2011
  • pp: 4740–4755
« Show journal navigation

Description of the modes governing the optical transmission through metal gratings

Mickaël Guillaumée, L. Andrea Dunbar, and Ross P. Stanley  »View Author Affiliations


Optics Express, Vol. 19, Issue 5, pp. 4740-4755 (2011)
http://dx.doi.org/10.1364/OE.19.004740


View Full Text Article

Acrobat PDF (1728 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

An analytical model based on a modal expansion method is developed to investigate the optical transmission through metal gratings. This model gives analytical expressions for the transmission as well as for the dispersion relations of the modes responsible for high transmission. These expressions are accurate even for real metals used in the visible – near-infrared wavelength range, where surface plasmon polaritons (SPP’s) are excited. The dispersion relations allow the nature of the modes to be assessed. We find that the transmission modes are hybrid between Fabry-Pérot like modes and SPP’s. It is also shown that it is important to consider different refractive indices above and below the gratings in order to determine the nature of the hybrid modes. These findings are important as they clarify the nature of the modes responsible for high transmission. It can also be useful as a design tool for metal gratings for various applications.

© 2011 OSA

1. Introduction

The observation of enhanced optical transmission (EOT) through a periodic array of subwavelength holes in a metal film was reported more than ten years ago by Ebbesen et al. [1

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

]. This observation has lead to many studies exploring the complex process responsible for EOT as well as its potential for various applications [2

2. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

]. To clarify the EOT process observed in Ref. [1

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

], many theoretical studies considered metal lamellar gratings, i.e. the one dimensional equivalent of a periodic array of holes. As pointed out in several studies [3

3. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000). [CrossRef]

,4

4. F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002). [CrossRef]

], care must be taken in the analogy between one and two dimensional structures. In the one dimensional case there is always a propagating mode inside slits for TM polarization (electric field perpendicular to the slit), whereas in the two dimensional case a cut-off wavelength exist for holes. Consequently, the transmission process for the two types of structures is different. However, even one dimensional gratings show complex transmission properties.

The exact role of SPP in the transmission process has been controversial. Several studies observed that the transmission at the SPP condition is close to zero [11

11. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

,12

12. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13(12), 4485–4491 (2005). [CrossRef] [PubMed]

]. Also, peaks observed in close proximity to the SPP condition were sometimes not attributed to SPP excitation but to the discontinuity observed when a diffracted order passes from evanescent to propagating [13

13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

,14

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(5), 490–495 (2007). [CrossRef]

], i.e. at the Rayleigh anomaly. We see from this brief résumé that the nature of the coupled modes responsible for the transmission peaks is not clear.

In the present work, an analytical model derived from the work of Lochbihler et al. [15

15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

] is developed in Sect. 2. From this analytical model, accurate analytical formula for the transmission and for the dispersion relations of the transmission resonances are obtained in Sect. 3. This allows the nature of the modes responsible for high transmission to be determined. Also, it is shown in Sect. 4 that the controversy regarding the role of the SPP modes in the transmission process can be lifted by considering asymmetric gratings (i.e. with a different refractive index above and below the grating) instead of symmetric ones (i.e. with the same refractive index above and below the grating). In light of this discussion, the symmetric grating is revisited in some details. Finally, the low transmission observed at the SPP condition is discussed.

2. Theoretical formalism

2.1 Structure under study

We study the transmission of a plane wave through an infinite metal lamellar grating of period p, thickness h and slit width w, see Fig. 1
Fig. 1 A schematic of the studied structure and definitions used in this article for the structure dimensions and incident field directions.
. The metal permittivity is denoted ε while the permittivity above the grating, inside the slits and below the grating is denoted ε 1, ε 2 and ε 3 respectively. The incident wave, with wavevector k 0 = 2π/λ, propagates in a direction contained in the (x, y) plane and forms an angle θ with the y axis. With these conditions, the grating does not mix polarization so the two different polarizations can be treated independently. Only TM polarization (magnetic field in the z direction) is considered as TE polarization (electric field in the z direction) does not excite SPP.

2.2 Theoretical development

The model used to calculate the optical properties of metal lamellar gratings are detailed here. As a starting point, the method of Lochbihler et al. [15

15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

] is considered. In this method, surface impedance boundary conditions (SIBC) are used at metal dielectric interfaces. SIBC have been used successfully for gold gratings in the visible to infrared wavelength range [16

16. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26(12), 2348–2354 (1987). [CrossRef] [PubMed]

,17

17. M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009). [CrossRef]

]. Using SIBC, the electromagnetic field in the metal is not calculated. Therefore, this model assumes there is no evanescent tunnelling through the metal and is limited to metal walls with thickness greater than the metal skin depth, as is the case in the present study.

To satisfy the continuity of tangential components of electric and magnetic fields, the continuity of Hz and its normal derivative are considered. Matching Hz at y = ± h/2, multiplying for any m = j the resulting equations by Xj(x) and integrating over the region 0 ≤ xw as done in Ref. [15

15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

], the following set of equations is obtained for any m = j:
Ij[ajexp(iβjh/2)+bjexp(iβjh/2)]=n=(rn+δn,0)Kj,n,
(4)
Ij[ajexp(iβjh/2)+bjexp(iβjh/2)]=n=tnKj,n,
(5)
where the overlap integrals are defined by:
Ij=0wXj(x)Xj(x)dx=[1+(η2μj)2]w2+η2μj2,
(6)
Kj,n=0wexp(ikx,nx)Xj(x)dx.
(7)
In a similar way, the normal derivative of Hz are matched at y = ± h/2 for 0 ≤ xw and SIBC is considered at metal - dielectric interfaces at y = ± h/2 for wxp. The obtained equations are multiplied for any n = q by exp(-ikx , qx) and integrated over the region 0xp. This gives the following set of equations for any n = q:
iky1,q(rqδq,0)=iε1ε2m=0βm[amexp(iβmh/2)bmexp(iβmh/2)]Jq,m+η1n=(rn+δn,0)Qq,n,
(8)
iky3,qtq=iε3ε2m=0βm[amexp(iβmh/2)bmexp(iβmh/2)]Jq,m+η3n=tnQq,n,
(9)
where ηd = -dk 0/ε 1/2 (d = 1 or 3) and the overlap integrals are defined by:
Jq,m=1/p0wexp(ikx,qx)Xm(x)dx,
(10)
Qq,n=1/p0wexp[i(kx,nkx,q)x]dx.
(11)
The calculation of the unknown coefficients rn, tn, am and bm can then be performed after truncation and writing Eqs. (4), (5), (8), (9) into a matrix equation.

In order to have a better understanding of the transmission properties of the grating, two simplifications are made here as compared to the model described in Ref. [15

15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

]: (i) The fundamental slit mode β 0 is the only one considered as it is the only propagating mode for narrow slits, the others being evanescent. This approximation has already been made for narrow slits (i.e. w < λ/10) to study light transmission through slit arrays [4

4. F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002). [CrossRef]

,11

11. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

,13

13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

,18

18. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973). [CrossRef]

,19

19. Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000). [CrossRef]

] but also hole arrays [20

20. L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008). [CrossRef]

]. (ii) The Qq , n matrix defined by Eq. (11), which represents the overlap between plane waves, is considered as the identity matrix. This approximation, also valid for narrow slits, means that the plane waves do not mix via scattering from the slits. With these two approximations, Eqs. (4), (5), (8), (9) lead to analytical expression of rn, tn, am and bm. The transmission amplitude of order q tq is expressed as:
tq=4(ε3/ε2)Jq,0K0,0(ky1,0+k0ε1/2)1(ky3,q+k0ε1/2)1ky1,0Y2(n=Y3,n+Y2)(n=Y1,n+Y2)eiβ0h(n=Y3,nY2)(n=Y1,nY2)eiβ0h,
(12)
where Yd , n = (εd/ε 2)Jn ,0 K 0, n(kyd , n + εdk 0 ε -1/2)−1 for d = 1 or 3 and Y 2 = I 0/β 0. The transmission through the grating is then obtained summing the transmission intensities Tq of any propagating diffracted orders where Tq=(ε1cosθout,q)/(ε3cosθ)|tq|2 with θout , q the angle of the q th outward propagating order.

2.3 Comparison with other models

3. Dispersion relation of transmission resonances

3.1 Derivation

3.2 Validity of the mode equation

Equation (13) reduces the transmission problem to a set of modes and gives the position of the transmission peaks. As an example, the total transmission of a gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = 1, ε 3 = 2.25 and normal incidence as a function of p and λ is plotted in Fig. 3(a)
Fig. 3 (a) Transmission of a gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = 1, ε 3 = 2.25 and normal incidence in function of p and λ. The brighter the region, the larger is the transmission. (b) Dispersion relations of the modes obtained from the solutions of Eq. (13) (blue and red dashed curves) for the same set of parameters as in Fig. 3(a). The cyan and green solid lines correspond to the dispersion relations of SPP modes excited at the top and bottom of the grating respectively.
. Note that plotting the transmission as a function of kx ,0 and λ would lead to similar observations. It is however more convenient to plot the transmission as a function of p and λ for an easier identification of the grating resonances which appear close to λ = (√εd)p/n (d = 1 or 3). For the same set of parameters as in Fig. 3(a), solutions to Eq. (13) are represented by the blue and red dashed curves in Fig. 3(b). The SPP dispersion relations for the top and bottom interfaces are represented by cyan and green solid curves. All transmission maxima observed in Fig. 3(a) match with the dispersion curves plotted in Fig. 3(b). This shows that the excitation of the modes whose dispersion relations are given by Eq. (13) leads to the transmission peaks.

4. Description of the modes responsible for high transmission

The nature of the modes responsible for the high transmission is determined in the present section. At first, the dispersion relations of the uncoupled modes are extracted from Eq. (13).

4.1 Periodicity related effects

Observing Fig. 3, one sees that high transmission is strongly dependant on period. The only terms dependant on period in Eq. (13) are the Yd , n terms. These terms present two critical points: (i) at Rayleigh anomalies, Yd , n terms are discontinuous; (ii) at SPP conditions, Yd , n terms present poles. Yd , n is rewritten here for clarity:
Yd,n=εdε2Jn,0K0,n(kyd,n+εdk0ε1/2).
(19)
Rayleigh anomalies occur when a diffracted order above or below the grating passes from evanescent to propagating, i.e. at kyd , n = 0. At this condition, kyd , n is discontinuous as it abruptly changes from an imaginary to a real value, this in turn causes a discontinuity of the Yd , n terms. At normal incidence, this condition is expressed as:
λRd,n=pnεd.
(20)
As an example, the term Y 1,1 is plotted in Fig. 4(a)
Fig. 4 Real (blue curves) and imaginary part (red curves) of the terms (a) Y 1,1; (b) ΣY 1,n (solid lines) and Y 2 (dashed lines) for a gold grating for w = 50 nm, h = 600 nm, p = 1000, ε 1 = ε 2 = 1 and normal incidence. In panel (a) λR 1,1 and λSPP 1,1 are represented by vertical dashed lines and the phase of Y 1,1 is represented by the violet curve and the right hand axis corresponds to the phase of Y 1.
for a gold grating with w = 50 nm, p = 1000 nm, ε 1 = ε 2 = 1 and normal incidence. It shows that Y 1,1 presents a discontinuity at λR 1,1, see the abrupt change in the phase of Y 1,1, arg(Y 1,1).

SPP modes are excited when kx , n = kSPPd where kSPPd is the SPP wavevector. Under SIBC, kSPPd = k 0[(εdεεd 2)/ε]1/2 [26

26. F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008). [CrossRef]

] and SPP modes are excited at:
λSPPd,n=pnεdεεdε.
(21)
The above equation corresponds to the condition at which the denominator of Yd , n in Eq. (19) is zero, hence producing a pole. This is what is observed in Fig. 4(a) where Y 1,1 presents a pole at λSPP 1,1. λRd , n and λSPPd , n appear each time in pair as they both depend linearly on period and incident angle. λSPPd , n is red shifted as compared to λRd , n. In the case of a PEC, λRd , n = λSPPd , n, i.e. both the resonance and the discontinuity due to the periodic structure appear under the same condition.

4.2 Fabry-Pérot resonances

Far from λSPPd , n, Yd , n terms are small as compared to Y 2. This is what is observed in Fig. 4(b). Solutions to Eq. (13) can then be approximated by wavelengths λFP , l which fulfill the condition:
β0h=πl,
(22)
where l is an integer. At these conditions, FP resonances are excited inside the slits. In analogy with waveguides, it will be shown later that the integer l corresponds to the number of field maxima inside the slits.

4.3 Coupled modes

The goal of this section is to determine the nature of the modes responsible for high transmission. To this aim, it is shown here that it is easier to consider asymmetric gratings than symmetric ones as the top and bottom SPP modes are well separated.

It was shown in Sect. 4.1 and 4.2 that both FP and SPP modes are contained in the dispersion relation given by Eq. (13). Also, Fig. 3 shows that the coupled modes are asymptotic to SPP modes at long wavelength. This agrees with the description made by Marquier et al. [10

10. F. Marquier, J. J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13(1), 70–76 (2005). [CrossRef] [PubMed]

] who described the modes of the grating as a coupling between FP and SPP modes. However, several past studies attributed the high transmission observed to the discontinuity produced by the Rayleigh anomaly instead of the excitation of SPP modes [11

11. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

,13

13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

,14

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(5), 490–495 (2007). [CrossRef]

]. A point which could have lead to some ambiguity regarding the respective role of SPP and Rayleigh anomaly is the fact that most of the past study considered symmetric environment, i.e. ε 1 = ε 3. The symmetric grating is a very particular case.

To show this, schematics of the mode coupling for three different types of gratings are shown in Fig. 5
Fig. 5 Schematic of the coupling mechanism between SPP and FP modes in metal gratings. In panel (a) and (b) ε 1 < ε 3 and the difference between ε 1 and ε 3 is reduced from (a) to (b). In panel (c), ε 1 = ε 3.
. Although these schematics could appear oversimplified as compared to the case of Fig. 3, it helps to give a general idea of the coupling mechanism. The system is composed of 3 types of modes, each one being represented in Fig. 5: (i) SPP modes excited above the grating λSPP 1, n (cyan lines); (ii) SPP modes excited below the grating λSPP 3, n (green lines); (iii) FP modes λFP , l (black lines). The coupling of these modes leads to new modes represented by red and blue lines. In Fig. 5(a), ε 1 < ε 3 and the top and bottom SPP modes are well separated. As the contrast between ε 1 and ε 3 is decreased, λSPP 1, n and λSPP 3, n become closer [cf. Fig. 5(b)] until λSPP 1, n = λSPP 3, n for ε 1 = ε 3. The top and bottom SPP modes couple together via the slits, which creates two degenerated SPP modes: a symmetric low frequency mode λSPP - , n and an antisymmetric high frequency mode λSPP + , n. This degeneracy of the SPP modes is discussed in the Appendix. The band structure of the symmetric case is then strongly modified as compared to the asymmetric one, see Fig. 5(c). For example, both λFP 1 and λFP 2 modes in Fig. 5(c) couple to λSPP ,1 as the period is increased. The mode corresponding to the coupling between λFP 2 and λSPP ,1 represented in red crosses the SPP line. This crossing could be interpreted as if this mode was not a hybrid FP-SPP mode. But this crossing occurs because we are in the presence of the two degenerated SPP modes λSPP - ,1 and λSPP + ,1. Increasing the contrast between ε 1 and ε 3 removes any ambiguity regarding the nature of the coupled modes. Note also that it is easier to determine graphically the nature of the coupled modes in a (p, λ) diagram as done in Fig. 3 rather than in a (λ, h) diagram as in Ref. [13

13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

].

5. The symmetric grating case

The transmission of a symmetric gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence is plotted in Fig. 6
Fig. 6 (a) Transmission of a gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence in function of p and λ. The brighter the region, the larger is the transmission. (b) Dispersion relations of the modes obtained from the solutions of Eq. (15) (blue dotted curves) and Eq. (16) (red dashed curves) for the same set of parameters as in Fig. 6(a). The green solid curves correspond to the SPP dispersion relations.
together with the mode dispersion. As the SPP modes excited at each interface appear at the same condition, there are half as many SPP lines as compared to the asymmetric case shown in Fig. 3. As a consequence, it is easier to distinguish the horizontal lines close to λFP , l.

5.1 High transmission far from SPP conditions

From the FP condition given by Eq. (22), high transmission far from SPP conditions is expected at λFP ,1 ≈1700 nm, λFP ,2 ≈845 nm and λFP ,3 ≈620 nm for the slit parameters of Fig. 6. Indeed, broad transmission maxima are observed in Fig. 6(a) close to these three wavelengths for any period except when in close proximity with λSPP 1, n. For example, for p ranging from 500 to 1500 nm, high transmission is observed close to λFP ,1.

Maps of the magnetic field intensity at each peak position close to λFP ,1 with l = 1, 2 and 3 are shown in Fig. 7(a)
Fig. 7 Map of the magnetic field intensity (a) - (c) and the magnetic field amplitude (d) – (f) plotted in the (x, y) plane over one period (p = 500 nm) in the x direction for different wavelengths. Each figure is obtained from the simplified model. Grey rectangles represent the metal regions.
7(c) for p = 500 nm. Each figure is plotted in the (x, y) plane over one period in the x direction. Grey rectangles represent the metal regions. In each case, the field is strongly localized inside the slit with l field maxima. This shows that the slit is acting as a FP cavity. When l field maxima are observed in the slit with l odd, the transmission peak position is predicted from Eq. (15), whereas it is obtained from Eq. (16) for l even. Plotting the field amplitude [Fig. 7(d)7(f)], one sees that the field is symmetric and antisymmetric with respect to the y = 0 axis for l odd and even respectively.

5.2 High transmission near SPP conditions

The different branches constituting the dispersion relations of symmetric gratings are labeled λ(n,l) in Fig. 6(b), i.e. poles for which λSPP 1, n +1 < λ < λSPP 1, n and λFP , l +1 < λ < λFP , l -1. Only the first five branches are labeled for clarity.

Focusing on the λ(0,1) branch, the mode is red-shifted as the period is increased. Mathematically, this is due to the resonance of Y 1,1 occurring at λSPP 1,1 which is red-shifted as the period is increased. The mode continuously evolves from a FP-like mode in the 500 - 1500 nm period range with nearly flat dispersion to an SPP-like one for periods ranging from 2000 to 3000 nm, where λ(0,1) is asymptotic to λSPP 1,1. Whereas the magnetic field intensity is mainly confined inside the slit for the λ(0,1) branch at small periods [see Fig. 7(a)], an intense field is present on the surface for larger periods, whilst a FP character remains inside the slit, see Fig. 8(a)
Fig. 8 Map of the magnetic field intensity (a), (c) and the magnetic field amplitude (b), (d) plotted as in Fig. 7 for λ = 3000 nm and two different periods.
for λ = 3000 nm and p = 2977 nm. Figure 8(b) shows that the magnetic field amplitude along λ(0,1) keeps a symmetric profile for large periods.

The λ(0,2) branch is also red-shifted as the period is increased. It crosses λSPP 1,1 and merges to the λ(1,0) branch which stays asymptotic to λSPP 1,1 for large periods. The two maxima of the FP cavity become less confined into the slit as p is increased along the λ(0,2) branch and afterwards the λ(1,0) branch. In Fig. 8(c), the field is no longer confined inside the slit for p = 2999.6 nm and λ = 3000 nm but it extends in the y direction above and below the grating. As shown in Fig. 8(d), the magnetic field amplitude of the λ(1,0) branch keeps the antisymmetric profile of the λ(0,2) branch.

At long wavelengths, the λ(0,1) and λ(1,0) modes present similarities with the symmetric and antisymmetric SPP modes observed with thin metal films. In both thin metal films and metal gratings, symmetric modes are red-shifted as compared to λSPP 1, n, whereas the antisymmetric modes are blue-shifted. Also, the antisymmetric modes correspond to the long range SPP’s, which give sharp resonances [see the λ(1,0) mode in Fig. 9], whereas the symmetric modes correspond to the short range SPP’s, presenting broader resonances [λ(0,1) mode in Fig. 9]. The difference here as compared to a flat metal film is that SPP’s are coupled on both sides of the film for any thickness due to the presence of slits.

As the two SPP modes couple together via the slits and creates two SPP modes, the first FP mode, which is symmetric, couples with the symmetric SPP mode of order n = 1. Similarly, the second FP mode, which is antisymmetric, couples with the antisymmetric SPP mode of order n = 1. Finally, it can be said that Eq. (15) and Eq. (16) give the dispersion relation of respectively symmetric and antisymmetric hybrid “FP-SPP” modes.

6. Minimum in transmission at the SPP condition

It has been shown previously that high transmission is induced by the excitation of hybrid “FP-SPP” modes. We want to clarify here what exactly happens at the SPP resonance.

At λ = λSPPd , n, Yd , n is resonant and one sees from Eq. (12) that the transmission amplitude tq is nearly zero for any propagating diffracted order; i.e. for q < n. Consequently, the total transmission is always minimal at λSPPd , n, as observed in Fig. 3 and Fig. 6. This is in agreement with Cao et al. who already observed that the transmission is nearly zero at the SPP condition [11

11. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

]. It also explains why transmission peaks are hardly observed in transmission spectra when the dispersion relations predict transmission resonances that are too close to the SPP conditions. Lalanne et al. investigated in a past study the zero observed at the SPP condition in terms of reflected and transmitted intensities of single interfaces [13

13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

]. They found that the transmitted intensity of the incident wave into the fundamental mode is close to zero at λSPPi , n. This means that there is no coupling into or out of the slit at the SPP resonance.

Physically, this low transmission can be understood as the fact that at the SPP condition, the field is bounded at the metal interface but not above the slit, as seen from Fig. 10
Fig. 10 Map of the magnetic field intensity plotted as in Fig. 7 for λ = 1700 nm and p = 1693 nm.
at the crossing of the λ(0,2) branch with λSPP 1,1 for p = 1693 nm and λ = 1700 nm. This inhibits light from coupling into the slit mode. Note that the degeneracy of the top and bottom SPP modes in the case of symmetric grating leads to two transmission minima. This is discussed in the Appendix.

7. Conclusion

8. Appendix: Coupled SPP modes and photonic bandgap

This section is dedicated to the degenerated SPP modes of symmetric gratings. It is shown that the degenerated SPP modes lead to two transmission minima. This should also be useful in order to avoid confusion with photonic bandgaps observed in the propagation of surface plasmon polaritons on corrugated surfaces [27

27. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

].

The degeneracy of the top and bottom SPP modes in the case of symmetric grating cannot be observed considering only one mode inside the slit. Due to symmetry reasons, it is necessary to consider at least three slit modes. Therefore, the model of Ref. [15

15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

]. is now used.

The transmission spectrum of a gold grating is plotted in Fig. 11(a)
Fig. 11 (a) Transmission spectrum of a gold grating for w = 600 nm, h = 600 nm, λ = 1750 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence. The magnetic field amplitudes at λSPP + ,1 = 1750.5 nm and λSPP - ,1 = 1751.9 nm are plotted in panel (b) and (c) respectively.
on a log scale for p = 1750 nm, w = 700 and h = 600 nm. Two minima, labeled λSPP + ,1 and λSPP - ,1, are observed between λR 1,1 ≈ 1750 nm and λSPP 1,1 ≈ 1757 nm. This shows that there is a coupling between the two SPP modes excited on both sides of a symmetric grating The amplitude of the magnetic field for the transmission minima λSPP + ,1 and λSPP - ,1 are plotted in Fig. 11(b) and 11(c) respectively. The field profile is similar to the one already shown in Fig. 10 at the SPP condition. The field appears saturated above the grating in Fig. 11 because the scale has been adjusted such that the weak field amplitude below the grating can be observed. The low frequency mode λSPP - , n presents a symmetric profile and the high frequency mode λSPP + , n a antisymmetric one. This is in agreement with what is said in Sect. 4.3.

It should be noted that the SPP splitting observed here occurs due to the finite grating thicknesses and coupling between the two grating surfaces. Consequently, the analogy with photonic crystals as done by Barnes et al. cannot be used for the splitting observed here [27

27. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

].

Acknowledgments

We would like to thank Fernando de León-Pérez, Luis Martín-Moreno, Gaël Nardin and Philippe Lalanne for fruitful discussions. This work was funded by the European Community, project no. IST-FP6- 034506 'PLEAS'.

References and links

1.

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

2.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]

3.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000). [CrossRef]

4.

F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002). [CrossRef]

5.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999). [CrossRef]

6.

S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63(3), 033107 (2001). [CrossRef]

7.

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000). [CrossRef]

8.

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86(24), 5601–5603 (2001). [CrossRef] [PubMed]

9.

D. Crouse and P. Keshavareddy, “Role of optical and surface plasmon modes in enhanced transmission and applications,” Opt. Express 13(20), 7760–7771 (2005). [CrossRef] [PubMed]

10.

F. Marquier, J. J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13(1), 70–76 (2005). [CrossRef] [PubMed]

11.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

12.

Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13(12), 4485–4491 (2005). [CrossRef] [PubMed]

13.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]

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(5), 490–495 (2007). [CrossRef]

15.

H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]

16.

R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26(12), 2348–2354 (1987). [CrossRef] [PubMed]

17.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009). [CrossRef]

18.

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973). [CrossRef]

19.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000). [CrossRef]

20.

L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008). [CrossRef]

21.

The commercially available software GSolver has been used. More information can be found at http://www.gsolver.com/ (2010).

22.

Data may be retrieved at http://www.sopra-sa.com (20010).

23.

The commercially available software Omnisim has been used. More information can be found at http://www.photond.com/products/omnisim.htm (2010).

24.

S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15(7), 4310–4320 (2007). [CrossRef] [PubMed]

25.

A. Yariv, Optical electronics in modern communications (Oxford University Press, 2007).

26.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008). [CrossRef]

27.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

28.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1988).

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Diffraction and Gratings

History
Original Manuscript: December 9, 2010
Revised Manuscript: January 21, 2011
Manuscript Accepted: February 8, 2011
Published: February 25, 2011

Citation
Mickaël Guillaumée, L. Andrea Dunbar, and Ross P. Stanley, "Description of the modes governing the optical transmission through metal gratings," Opt. Express 19, 4740-4755 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4740


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
  2. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007). [CrossRef] [PubMed]
  3. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000). [CrossRef]
  4. F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002). [CrossRef]
  5. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999). [CrossRef]
  6. S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63(3), 033107 (2001). [CrossRef]
  7. S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000). [CrossRef]
  8. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86(24), 5601–5603 (2001). [CrossRef] [PubMed]
  9. D. Crouse and P. Keshavareddy, “Role of optical and surface plasmon modes in enhanced transmission and applications,” Opt. Express 13(20), 7760–7771 (2005). [CrossRef] [PubMed]
  10. F. Marquier, J. J. Greffet, S. Collin, F. Pardo, and J. L. Pelouard, “Resonant transmission through a metallic film due to coupled modes,” Opt. Express 13(1), 70–76 (2005). [CrossRef] [PubMed]
  11. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]
  12. Y. Xie, A. Zakharian, J. Moloney, and M. Mansuripur, “Transmission of light through a periodic array of slits in a thick metallic film,” Opt. Express 13(12), 4485–4491 (2005). [CrossRef] [PubMed]
  13. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003). [CrossRef]
  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(5), 490–495 (2007). [CrossRef]
  15. H. Lochbihler and R. A. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32(19), 3459–3465 (1993). [CrossRef] [PubMed]
  16. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26(12), 2348–2354 (1987). [CrossRef] [PubMed]
  17. M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009). [CrossRef]
  18. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973). [CrossRef]
  19. Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000). [CrossRef]
  20. L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008). [CrossRef]
  21. The commercially available software GSolver has been used. More information can be found at http://www.gsolver.com/ (2010).
  22. Data may be retrieved at http://www.sopra-sa.com (20010).
  23. The commercially available software Omnisim has been used. More information can be found at http://www.photond.com/products/omnisim.htm (2010).
  24. S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15(7), 4310–4320 (2007). [CrossRef] [PubMed]
  25. A. Yariv, Optical electronics in modern communications (Oxford University Press, 2007).
  26. F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008). [CrossRef]
  27. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]
  28. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1988).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited