## Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes

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

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

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]

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

*model*) is proposed. The

*equivalent*circuit will implicitly contain all the information already provided by the analytical solution, but it has the additional advantage of making its physical interpretation much easier. The role of the different waveguide modes in the onset of ET will be analyzed and connected with the frequency dependence of the different elements of the proposed equivalent circuit. The same analysis will also be applied to elucidate the role of surface waves (or

*spoof plasmons*) in the onset of ET. Finally the present proposal will allow us to link the reported results —which come basically from a diffraction theory analysis— to the circuit theory approach proposed in [11, 12

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]

*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

*E*at

_{y}*z*= 0

^{+}can be expanded into the following Fourier series:

*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

*f*(

_{nm}*x*,

*y*) = cos(2

*nπx*/

*a*)cos(2

*mπy*/

*a*) . Using waveguide theory [16], the electric field component

*E*can be written as

_{x}*g*(

_{nm}*x*,

*y*) = sin(2

*nπx*/

*a*)sin(2

*mπy*/

*a*).

*f*. However, for small holes and not very large values of

_{nm}*n*and

*m*(taking into account that

*E*must be zero at the metallic screen), the following approximation applies [15

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

*w*and

*h*stands for the waveguide and hole sections respectively, and

*a*

^{2}is the waveguide section. Thus, it is finally found that

*E*should be almost zero at the hole (and zero on the metallic screen). Therefore, from (2) and (4):

_{x}*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], that is,

*H*= −

_{x}*Y*

_{0}= − √ε

_{0}/

*μ*

_{0}and

*H*= 0

_{y}^{1}. 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:

*λ*=

_{n}*a*/

*n*, the upper limits of the series in (6) can be determined by imposing a “resolution” equal to the hole size

*b*. This leads to

*N*,

*M*≈

*a*/

*b*, which completes the determination of

*T*from (6). Results for the transmission coefficient for several values of

*a*/

*b*computed from (6) are shown in Fig.2 together with data coming from full-wave electromagnetic simulations using the commercial software

*CST Microwave Studio*. Both set of results agree quite well not only qualitatively but also quantitatively. The figure also shows other previous analytical results on the same structure [15

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

*ω*= 2

*πf*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], a direct comparison between (9) and (6) leads to the following expressions for the capacitive,

*B*, and inductive,

_{C}*B*, susceptances appearing in (9

_{L}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]

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]

*a*(in our case, it also corresponds to the cutoff frequency of the TM

_{02}mode), the admittance of the TM

_{02}mode suddenly grows to infinity, which makes the term associated with this latter mode be dominant in the capacitive susceptance series (10). Under the same circumstances, however, the admittance of the TE

_{20}mode goes to zero, which means that there is no singularity in the inductive susceptance BL. As it is well know from Bethe’s theory, for normal incidence and frequencies far and below Wood’s anomaly, a small hole has an inductive behavior (it can be modeled by an equivalent magnetic dipole). Nevertheless, as frequency approaches Wood’s anomaly, we have just seen that the absolute value of the associated capacitive susceptance BC grows to infinity and, at certain frequency, it will cancel out the inductive susceptance associated with the hole (namely,

*B*+

_{C}*B*= 0) and will give rise to total transmission. It is also interesting to note that, within the same previous frequency range, all admittances except

_{L}*Y*

_{0,2}

^{TM}have a smooth frequency dependence. In that case, the inductive susceptance

*B*(11) is found to be roughly proportional to (

_{L}*a*/

*b*)

^{2}. However, the capacitive susceptance, which is dominated by

*Y*

_{0,2}

^{TM}, will be proportional to

*a*/

*b*. This means that |

*B*|/

_{C}*B*∝

_{L}*b*/

*a*as λ →

*a*, which implies that the smaller the hole, the smaller the absolute value of

*B*is with regard to

_{C}*B*. In other words, the smaller the hole, the closer ET is to Wood’s anomaly. An additional observation can be made after considering that the absolute value of the capacitive susceptance still increases for frequencies above ET until it becomes infinity at Wood’s anomaly. At this last frequency, the

_{L}*LC*tank in the equivalent circuit of Fig.1(e) becomes a short circuit and total reflection will appear. Therefore the equivalent circuit of Fig.1(e) along with the transformations (10) and (11) explains satisfactorily both ET and Wood’s anomaly in periodically perforated zero thickness screens.

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]

*R*

_{0}= 1/

*Y*

_{0}(=

*Z*

_{0}) = √

*μ*

_{0}/

*ε*

_{0}≈ 377Ω. The frequency of excitation of surface plasmons with the appropriate wavevector,

*k*= 2π/

*a*, can now be identified with the frequency of resonance of the

*loaded LC*resonator shown in Fig. 3. This frequency of resonance,

*ω*′ –

*iω*″, must be complex in order to account for radiation losses, and can be computed as the solution to the following implicit equation:

*k*= 2

*π*/

*a*), and its imaginary part gives the

*lifetime*of the wave through

*τ*= 1/

*ω*″. Clearly, if

*R*

_{0}is much larger than |

*B*| as well as

_{C}*B*, 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

_{L}*a*/

*b*is, the closer both frequencies appear.

*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.

*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 TE

_{10}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 TE

_{10}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 TE

_{10}mode:

*Y*

_{TE10}=

*iY*

_{0}√(

*λ*/2

*b*)

^{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

*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.

*t*=

*a*/7 are shown and compared with 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]

## Acknowledgments

## Footnotes

1It is worth to mention that a completely different value for H and _{x}H was assumed in [15_{y}15. R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A |

## 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) |

2. | H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. |

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 |

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. |

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. |

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. |

7. | J. A. Porto, F. J. García-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. |

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 |

9. | J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

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. |

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. |

13. | C. Genet and T.W. Ebbesen, “Light in tiny holes,” Nature , |

14. | F.J. García-de-Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. , |

15. | R. Gordon, “Bethe’s aperture theory for arrays,” Phys. Rev. A |

16. | J. D. Jackson, |

**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

- 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]
- H. A. Bethe,"Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. B. Pendry, L. Martın-Moreno, and F. J. Garcıa-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- 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]
- 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.
- 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]
- C. Genet and T. W. Ebbesen, "Light in tiny holes," Nature 445, 39-46 (2007). [CrossRef] [PubMed]
- F. J. Garcıa-de-Abajo, "Colloquium: Light scattering by particle and hole arrays," Rev. Mod. Phys. 79, 1267-1290 (2007). [CrossRef]
- R. Gordon, "Bethe’s aperture theory for arrays," Phys. Rev. A 76, 053806 (2007). [CrossRef]
- J. D. Jackson, Classical Electrodynamics, Edt. (Wiley, New York, 1999), 3rd Ed.

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