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Optics Express

  • Vol. 17, Iss. 7 — Mar. 30, 2009
  • pp: 5571–5579
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Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes

R. Marqués, F. Mesa, L. Jelinek, and F. Medina  »View Author Affiliations


Optics Express, Vol. 17, Issue 7, pp. 5571-5579 (2009)
http://dx.doi.org/10.1364/OE.17.005571


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Abstract

In this letter, the problem of extraordinary (ET) transmission of electromagnetic waves through opaque screens perforated with subwavelength holes is addressed from an analytical point of view. Our purpose was to find a closed-form expression for the transmission coefficient in a simple case in order to explore and clarify, as much as possible, the physical background of the phenomenon. The solution of this canonical example, apart from matching quite well with numerical simulations given by commercial solvers, has provided new insight in extraordinary transmission as well as Wood’s anomaly. Thus, our analysis has revealed that one of the key factors behind ET is the continuous increase of excess electric energy around the holes as the frequency approaches the onset of some of the higher-order modes associated with the periodicity of the screen. The same analysis also helps to clarify the role of surface modes –or spoof plasmons–in the onset of ET.

© 2009 Optical Society of America

1. Introduction

Fig. 1. Perfect conductor screen perforated with square holes: front view (a) and two lateral cuts (b). Front (c) and lateral (d) views of the structure unit cell or equivalent waveguide. (e) Equivalent circuit for the discontinuity in the waveguide. It has been assumed that t → 0.

The structure under study is shown in Figs. 1(a)-(b). For normal incidence of a y-polarized electromagnetic plane wave, this structure is equivalent to a TEM waveguide with perfect electric conducting plates at the upper and bottom interfaces, perfect magnetic conducting plates at both lateral sides, and a square diaphragm located, say, at z = 0 (see Figs. 1(c)-(d)). Due to the symmetry of this structure, and assuming an incident field of amplitude equal to unity, the field component Ey at z = 0+ can be expanded into the following Fourier series:

Ey(x,y)=T+n=1NAn0TEfn0(x,y)+m=1MA0mTMf0m(x,y)+n,m=1N,M(AnmTE+AnmTM)fnm(x,y),
(1)

where T is the transmission coefficient, A TE, A TM are the coefficients of the (below cutoff) TE and TM waveguide modes excited at the discontinuity, and fnm(x,y) = cos(2nπx/a)cos(2mπy/a) . Using waveguide theory [16

16. J. D. Jackson, Classical Electrodynamics, Edt. Wiley, New York (1999), 3rd Ed.

], the electric field component Ex can be written as

Ex(x,y)=n,m=1(mnAnmTEnmAnmTM)gnm(x,y)
(2)

with gnm(x,y) = sin(2nπx/a)sin(2mπy/a).

The coefficients of the expansion (1) can be obtained from the orthogonality properties of functions fnm. However, for small holes and not very large values of n and m (taking into account that Ey must be zero at the metallic screen), the following approximation applies [15

15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

]:

wEyfnmdxdy=hEyfnmdxdyhEydxdy=wEydxdy=a2T,
(3)

where subindex w and h stands for the waveguide and hole sections respectively, and a 2 is the waveguide section. Thus, it is finally found that

An0TE2T;A0mTM2T;AnmTE+AnmTM4T.
(4)

For small holes, Ex should be almost zero at the hole (and zero on the metallic screen). Therefore, from (2) and (4):

AnmTE4Tn2n2+m2;AnmTM4Tm2n2+m2.
(5)

The transmission coefficient T can now be obtained after imposing the appropriate boundary conditions for the transverse magnetic field. Since the scattered field is produced by the electric currents induced in the diffraction screen, which are confined to the z = 0 plane, it is deduced from symmetry that all the tangential components of the scattered magnetic field must vanish at the aperture. This conclusion comes out from the fact that such induced currents are vectors invariant by reflection in the z = 0 plane, whereas the scattered magnetic field is a pseudo-vector, whose tangential components must change of sign after reflection in such plane. Therefore, the total tangential magnetic field in the hole must be equal to the incident field [16

16. J. D. Jackson, Classical Electrodynamics, Edt. Wiley, New York (1999), 3rd Ed.

], that is, Hx = −Y 0 = − √ε0/μ 0 and Hy = 01. Once the tangential magnetic field at the aperture has been evaluated, upon substitution of (4)-(5) in (1) the transmission coefficient T can be obtained after solving the following equation:

b2Y0(1T)=abTn=1N2nπY2n,0TEsin(nπba)+abTm=1M2mπY0,2mTMsin(mπba)
+a2Tn,m=1N,M4nmπ2(Y2n,2mTEn2n2+m2+Y2n,2mTMm2n2+m2)sin(nπba)sin(mπba),
(6)

Y2n,2mTE=iY0(nλa)2+(mλa)21
(7)
Y2n,2mTM=iY0/(nλa)2+(mλa)21.
(8)

Fig. 2. Transmission coefficient of the structure shown in Fig. 1 for different values of the ratio a/b versus the ratio (fWf)/fW, where fW = c/a is the Wood’s anomaly frequency, with c being the light velocity in free space. Solid Lines correspond to data from (6). Dotted lines correspond to data from CST. For comparison purposes, the numerical value for the ET frequency provided in [15] for a/b = 7.07 (i.e. holes covering a 2% of the total area) is shown with an arrow

Now, the equivalent circuit shown in Fig. 1(e) is proposed for the waveguide discontinuity problem shown in Figs. 1(c) and (d). The transmission coefficient for this equivalent circuit configuration can be found from the solution of the following equation:

Y0(1T)=T(iωC21iω2L)
(9)

where ω = 2πf is the angular frequency. The above equation clearly shows that total transmission is obtained at frequency ω 0 = √1/(LC). Considering now that the evanescent TE(TM) mode admittances are imaginary and positive(negative) [16

16. J. D. Jackson, Classical Electrodynamics, Edt. Wiley, New York (1999), 3rd Ed.

], a direct comparison between (9) and (6) leads to the following expressions for the capacitive, BC, and inductive, BL, susceptances appearing in (9

9. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

):

BC=ωC/2=1i{abm=1M2Y0,2mTMmπsin(mπba)
+(ab)2n,m=1N,M4Y2n,2mTMnmπ2m2n2+m2sin(nπba)sin(mπba)}
(10)
BL=12ωL=1i{abn=1M2Y0,2mTEnπsin(nπba)
+(ab)2n,m=1N,M4Y2n,2mTEnmπ2m2n2+m2sin(nπba)sin(mπba)}.
(11)

Fig. 3. Equivalent circuit for the computation of the frequency of excitation of surface waves with k = 2π/a.

iωC(ω)1iωL(ω)+2R0=0
(12)

(note that both C and L depends on frequency via the TM and TE admittances). The real part of the complex frequency is actually the frequency of excitation of the surface wave (for k = 2π/a), and its imaginary part gives the lifetime of the wave through τ = 1/ω″. Clearly, if R 0 is much larger than |BC| as well as BL, the frequency of excitation of the surface waves will be very close to the frequency of ET (although both frequencies will never coincide). Table 1 shows a comparison between these resonance frequencies and the ET frequencies for the cases analyzed in Fig. 2. This table shows that the higher the ratio a/b is, the closer both frequencies appear.

Table 1. Normalized resonance and extraordinary transmission frequencies, (fWf)/fW, for the cases studied in Fig. 2.

table-icon
View This Table

In fact, both frequencies have more than five identical significant digits for a/b > 5. Also, the imaginary part of ω′ becomes almost negligible for high values of this ratio, which shows that this frequency exactly coincides with the frequency of ET in the limit of very small holes (a/b → ∞). However, for smaller values of this ratio, both frequencies, although close, show a significant deviation. Accordingly, the imaginary part of ω′ becomes more significant as the ratio a/b increases. Larger holes would yield even higher differences between the frequencies associated with the surface plasmon and the extraordinary transmission.

Until now we have considered infinitely thin screens. However, the proposed theory can easily be extended to diffraction screens of finite thickness t. In this case the circuit model of Fig. 1(c) must be modified in order to include the evanescent waveguide formed by the hole. If the hole is small, it will be assumed that only the dominant TE10 mode is significantly excited inside the hole, and hence the effect of higher order modes are neglected. Thus the hole is modeled as an evanescent transmission line with admittance equal to the admittance of this TE10 mode. For square holes, this admittance (defined as the average current through the hole divided by the average voltage accross the hole) coincides with the wave admittance of the aformentioned TE10 mode: Y TE10 = iY 0 √(λ/2b)2 – 1. Moreover, only a fraction of the current flowing through the diffraction screen will go through the holes. This fraction can be roughly estimated as the fractional length along the x-axis (the axis perpendicular to the current) occupied by the holes; namely, b/a. Thus, from power conservation, the admittance Y′ seen in the diffraction screen at the input of the hole can be obtained from

P=12I2Y'=12(YTE10)1baI2
(13)

which gives Y' = (a/b)2 Y TE10. The circuit element providing this admittance transformation is an ideal transformer with turns ratio n = a/b. Therefore, this ideal transformer must be included between the resonant tank modeling the step discontinuity and the transmission line modeling the hole. The resulting equivalent circuit is shown in Fig. 4.

Fig. 4. Equivalent circuit for the structure of Figs. 1(a)-(b) with finite thickness (t ≠ 0).

In Fig.5 the computed results for a screen of thickness t = a/7 are shown and compared with
Fig. 5. Transmission coefficient of the structure shown in Fig.1 with b = a/6 and t = 0, a/7.
the results obtained for a zero-thickness screen. For comparison purposes the results obtained from numerical simulation using CST Microwave Studio are also shown. As it can be seen, there is a good qualitative agreement between theory and simulations. This agreement includes the presence of two transmission peaks, a well known effect in moderate thicknes screens (see [14

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

] and references therein). Taking into account the logaritmic scale, the quantitative agreement between theory and simulations is also quite good (more than four digits in the frequency of resonance). The source of the small numerical disagreement coul be attributed to the assumed approximate value of the transformer ratio.

In summary, an analytical solution for ET through thin diffraction screens has been presented. Our analysis, based on the equivalence with a waveguide discontinuity problem, shows that ET and Wood’s anomaly can both be explained from the peculiar behavior of the evanescent TM02 mode excited at the holes. Since this behavior is imposed by the screen periodicity, the analysis shows that it is this periodicity, rather than the physical nature of the screen, which is on the grounds of ET. Our analysis is also in agreement with the circuit theory of ET recently proposed by some of the authors. The analysis can also be applied to elucidate the role played by surface waves (or spoof plasmons) in ET. It has been shown that a radiating surface wave can be excited at a frequency very close to ET frequency but not exactly at the same frequency. This result suggests that radiating surface waves can play a significant role in the transitory states at the onset and at the end of a monochromatic ET steady state. Finally, the analysis was extended to diffraction screens of finite thickness, thus showing that the present theory also applies to ET in thick screens.

Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (projects TEC2007-65376, TEC2007-68013-C02-01, and CSD2008-00066), and by Junta de Andalucía (project TIC–253).

Footnotes

1It is worth to mention that a completely different value for Hx and Hy was assumed in [15

15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

], which is probably the sources of the numerical discrepancies with our analysis (see Fig. 2).

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 (London) 391, 667–669 (1998). [CrossRef]

2.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944). [CrossRef]

3.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58,, 6779–6782 (1998). [CrossRef]

4.

D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, “Crucial role of metal surface in enhanced transmission through subwavelength apertures,” Appl. Phys. Lett. 77, 1569–1571 (2000). [CrossRef]

5.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001). [CrossRef] [PubMed]

6.

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett. 29, 2500–2502 (2004). [CrossRef] [PubMed]

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]

8.

F. J. García-de-Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor sub-wavelength hole arrays,” Phys. Rev. E 72, 016608 (2005). [CrossRef]

9.

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

10.

A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cutoff,” Phys. Rev. Lett. 96, 073904 (2006) [CrossRef] [PubMed]

11.

F. Medina, F. Mesa, and R. Marqués, “Equivalent circuit model to explain extraordinary transmission,” in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA , 213–216, June 2008.

12.

F. Medina, F. Mesa, and R. Marqués, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microwave Theory Tech. 56, 3108–3120, 2008. [CrossRef]

13.

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

14.

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

15.

R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A 76, 053806 (2007). [CrossRef]

16.

J. D. Jackson, Classical Electrodynamics, Edt. Wiley, New York (1999), 3rd Ed.

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: January 16, 2009
Revised Manuscript: March 11, 2009
Manuscript Accepted: March 13, 2009
Published: March 24, 2009

Citation
R. Marquès, F. Mesa, L. Jelinek, and F. Medina, "Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes," Opt. Express 17, 5571-5579 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5571


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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 (London) 391, 667-669 (1998). [CrossRef]
  2. H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944). [CrossRef]
  3. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998). [CrossRef]
  4. D. E. Grupp, H. J. Lezec, T. W. Ebbesen, K. M. Pellerin, and T. Thio, "Crucial role of metal surface in enhanced transmission through subwavelength apertures," Appl. Phys. Lett. 77, 1569-1571 (2000). [CrossRef]
  5. L. Martın-Moreno, F. J. Garc’ıa-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, D. E. Grupp, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001). [CrossRef] [PubMed]
  6. M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Mart’ın-Moreno, J. Bravo-Abad, and F. J. Garcıa-Vidal, "Enhanced millimeter-wave transmission through subwavelength hole arrays," Opt. Lett. 29, 2500-2502 (2004). [CrossRef] [PubMed]
  7. J. A. Porto, F. J. Garcıa-Vidal, and J. B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999). [CrossRef]
  8. F. J. Garcıa-de-Abajo, R. Gomez-Medina, and J. J. Saenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72, 016608 (2005). [CrossRef]
  9. J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
  10. A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cutoff," Phys. Rev. Lett. 96, 073904 (2006) [CrossRef] [PubMed]
  11. F. Medina, F. Mesa, and R. Marques, "Equivalent circuit model to explain extraordinary transmission," in IEEE MTT-S Int. Microw. Symp. Dig., Atlanta, GA, 213-216, June 2008.
  12. F. Medina, F. Mesa, and R. Marques, "Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective," IEEE Trans. Microwave Theory Tech. 56, 3108-3120 2008. [CrossRef]
  13. C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007). [CrossRef] [PubMed]
  14. F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]
  15. R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007). [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, Edt. (Wiley, New York, 1999), 3rd Ed.

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