## Extraordinary transmission through 1, 2 and 3 holes in a perfect conductor, modelled by a mode expansion technique

Optics Express, Vol. 14, Issue 7, pp. 2552-2572 (2006)

http://dx.doi.org/10.1364/OE.14.002552

Acrobat PDF (360 KB)

### Abstract

We discuss a mode expansion technique to rigorously model the diffraction from three-dimensional pits and holes in a perfectly conducting layer with finite thickness. On the basis of our simulations we predict extraordinary transmission through a single hole, caused by the Fabry-Perot effect inside the hole. Furthermore, we study the fundamental building block for extraordinary transmission through hole arrays: two and three holes. Coupled electromagnetic surface waves, the perfect conductor equivalent of a surface plasmon, are found to play a key role in the mutual interaction between two or three holes.

© 2006 Optical Society of America

## 1. Introduction

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

02. J. Meixner and W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vol-lkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm,” Annalen der Physik **7**, 157–168 (1950). [CrossRef]

03. C. Flammer, “The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions,” J. Appl. Phys. **24**, 1218–1223 (1953). [CrossRef]

04. C. Flammer, “The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems,” J. Appl. Phys. **24**, 1224–1231 (1953). [CrossRef]

05. C.J. Bouwkamp, “Diffraction theory,” Reports on progress in physics **17**, 35–100 (1954). [CrossRef]

06. 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**, 667–669 (1998). [CrossRef]

07. H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a subwavelength slit: Waveg-uiding and optical vortices,” Phys. Rev. E **67**, 036,608 (2003). [CrossRef]

08. J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, “Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit,” Phys. Rev. E **69**, 026,601 (2004). [CrossRef]

09. H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. **94**, 053,901 (2005). [CrossRef] [PubMed]

10. M.G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A **3**, 1780–1787 (1986). [CrossRef]

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

12. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057,403 (2002). [CrossRef] [PubMed]

13. L. Martin-Moreno and F.J. Garcia-Vidal, “Optical transmission through circular hole arrays in optically thick metal films,” Opt. Express **12**, 3619–3628 (2004). [CrossRef]

14. S.H. Chang, S.K. Gray, and G.C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express **13**, 3150–3165 (2005). [CrossRef] [PubMed]

15. F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. **95**, 103,901 (2005). [CrossRef] [PubMed]

16. F.J. Garcia de Abajo, “Light transmission through a single cylindrical hole in a metallic film,” Opt. Express **10**, 1475–1484 (2002). [PubMed]

17. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A **4**, 1970–1983 (1987). [CrossRef]

15. F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. **95**, 103,901 (2005). [CrossRef] [PubMed]

18. J.M. Brok and H.P Urbach, “A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording,” J. Mod. Opt. **51**, 2059–2077 (2004). [CrossRef]

19. J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**, 847 (2004). [CrossRef] [PubMed]

20. A.P Hibbins, B.R. Evans, and J.R. Sambles, “Experimental verification of designer surface plasmons,” Science **308**, 670 (2005). [CrossRef] [PubMed]

## 2. Problem definition and system parameters

*x*,

*y*,

*z*) be a rectangular Cartesian coordinate system. Perpendicular to the

*z*-axis we have a perfectly conducting layer with finite thickness

*D*. In this layer a finite number of rectangular holes and pits are present. See Fig. 1. A hole is a rectangular cylinder that is open on both sides and is as long as the thickness of the layer; a pit has an open end at one side, either at

*z*=

*D*/2 or at

*z*= -

*D*/2, and a depth

*d*

_{p}<

*D*. The sub- or superscript

*p*denotes the number of the pit or hole. The lengths in the

*x*- and

*y*-direction are

*p*is given by Ω

_{p}= {(

*x*,

*y*)|

*x*<

*y*<

*z*>

*D*/2 and

*z*< -

*D*/2 are filled with homogeneous dielectrics with index of refraction

*n*

_{u}and

*n*

_{l}, respectively. Every hole and pit is filled with a homegeneous dielectric with index of refraction

*n*

_{p}. The corresponding relative permeabilities are

*ϵ*

_{u}=

*ϵl*=

*ϵ*

_{p}=

*μ*

_{0}everywhere.

*λ*. The local wavelengths are

*λ*

_{u}=

*λ*/

*n*

_{u},

*λ*

_{l}=

*λ*/

*n*

_{l}and

*λ*

_{p}=

*λ*/

*n*

_{p}. The corresponding wave vectors are

*k*

_{u}= 2

*π*/

*λ*

_{u},

*k*

_{l}= 2

*π*/

*λ*

_{l}and

*k*

_{p}= 2

*π*/

*λ*

_{p}. The harmonic time dependence of the electromagnetic field is given by the factor exp(-

*iωt*), with

*ω*< 0, which will be omitted throughout.

## 3. Mode expansions

### 3.1. Inside the holes and pits

*γ*

_{x}and

*γ*

_{y}determine the spatial behaviour in

*x*and

*y*-direction:

*m*

_{x}and

*m*

_{y}integers. The bold subscript

**α**= (

*α*

_{1},

*α*

_{2},

*α*

_{3},

*α*

_{4}) is a multi-index that describes four discrete variables:

*α*

_{1}(or

*p*) denotes the pit number,

*α*

_{2}indicates the polarization (TE or TM),

*α*

_{3}is determined by

*m*

_{x}and

*m*

_{y}and

*α*

_{4}specifies whether the mode is travelling upwards or downwards.

*z*= ±

*D*/2 only involve the

*x*- and

*y*- component of the fields, it is convenient to introduce the following notation:

*î*_{z}the unit vector in the

*z*-direction. In this way, the lower case

**e**and

**h**are the rotated transverse components of the electric and magnetic field. Furthermore, we split the transverse components of the modes into a real part that depends on

*x*and

*y*and a complex part that depends on

*z*:

*= (*

**α**̄*α*

_{1},

*α*

_{2},

*α*

_{3}) and thus the transverse vectorfield

**do not depend on the direction of propagation of the mode.**

*v*_{α}̄^{*}denotes complex conjugation [23]. Furthermore, the modes are orthogonal such that for different modes

**and**

*α*̄**′:**

*α*̄*z*-direction of a mode is given by:

**|**

*να*̄**〉**

*να*̄_{Ωp}a of a mode

**with itself is proportional to the flow of energy of this mode through a plane of constant**

*α*̄*z*.

*,*

*E*_{α}**H**) are complete in the following sense: any time harmonic electromagnetic field with frequency

_{α}*ω*satisfying the source-free Maxwell equations inside the holes and pits can be expressed as a linear combination of these mode functions. Hence, for

*z*between

*D*/2 and -

*D*/2, we have:

*a*

_{α}that will be determined by matching the field inside the holes and pits to the field above and below the conducting layer.

### 3.2. Above and below the layer

*b*are still to be determined and (

_{β}*,*

**E**_{β}**H**are plane waves with wave vector

_{β}**k**

_{u}above the layer and

*k*

_{l}below the layer. The transverse components (

*k*

_{x},

*k*

_{y}) of the wave vector are real and the

*z*-component is given by:

*iωt*) and from the fact that the scattered field propagates away from the conducting layer. The subscript

**= (**

*β**β*

_{1},

*β*

_{2}) is a short notation for the polarization (

*β*

_{1}) and the

*x*- and

*y*-component of the wave vector (

*β*

_{2}= (

*k*

_{x},

*k*

_{y})). The polarization can either be S or P. S-polarized means that the

*z*-component of the electric field is zero (and thus corresponds to TE polarization inside the holes and pits), while for P-polarization the z-component of the magnetic field is zero (TM polarization). Note that the integral ∫d

*β*

_{2}is a shorthand notation for ∫∫d

*k*

_{x}d

*k*

_{y}.

*x*and

*y*and a part that depends on

*z*:

**such that:**

*ν*_{β}*δ*is the Kronecker delta and the second is the two-dimensional Dirac delta function:

*ω*in the half spaces

*z*>

*D*/2 and

*z*< -

*D*/2, that propagates away from the conducting layer can be expanded in terms of the plane waves (

*,*

**E**_{β}*). In particular, we have, for the scattered transverse electric field:*

**H**_{β}*β*

_{2}is a short-hand notation for integrating over

*k*

_{x}and

*k*

_{y}.

**f**: ℝ

^{2}→ ℂ

^{2}:

*β*

_{1}= S or

*β*

_{1}= P. This operator is basically the integral version of the operator

*ω*

^{2}

*ϵϵ*

_{0}

*μ*

_{0}. This is, of course, the fingerprint of the

*coupled electromagnetic surface wave*. Although the integrand is integrable, in the numerical implementation prudence is necessary.

*z*is constant and in particular for

*z*= ±

*D*/2, the scattered transverse magnetic field can now be expressed in terms of the electric field:

*x*,

*y*) with -∞ <

*x*,

*y*< ∞.

## 4. Matching at the interfaces

*z*= ±

*D*/2, we have the following relations for the tangential electric and the tangential magnetic field:

_{α1}is, as before, the cross-section of the pit or hole that is denoted by index

*α*

_{1}. In Eq. (20a), because the layer is perfectly conducting, the sum of the incident and reflected tangential electric field vanishes at

*z*= ±

*D*/2, hence:

*x*,

*y*,±

*D*/2) within the holes and pits. The waveguide modes that constitute

**e**

^{pit}and

**h**

^{pit}vanish outside the pits and holes, as indicated by the rectangle function ∏ in Eq. (31) and (32) in Appendix B.

*ν*_{α¯ ′}by using the scalar product defined in Eq. (34):

*α*

_{4}is a summation overthe two directions of propagation (

*α*

_{4}is not contained in

*). This equation is valid for all*

**α**̄*′ hence for all*

**α**̄*α*

_{1}(counting the number of holes and pits), for all

*α*

_{2}(TE and TM polarization) and for all

*α*

_{3}(the mode numbers,

*m*

_{x}and

*m*

_{y}). Consequently, solving the system of Eq. (23) for all

*′ and for*

**α**̄*z*= ±

*D*/2 gives the waveguide mode expansion coefficients

*a*

*. Note that the term on the right acts as a source term. The factor 〈A (*

_{α}

*ν**)|*

**α**̄

*ν**′〉*

**α**̄_{Ωp′}is called the interaction integral. Physically speaking, it describes the interaction of a waveguide mode

*, via the scattered plane waves through operator A, with another mode*

**α**̄*′. In Appendix C we will discuss some of its properties and a method to compute it numerically.*

**α**̄**e**

_{β′}:

*r*

^{-1/3}, where

*r*is the distance to the edge. The field components parallel to the edge remain finite. Furthermore, the charge density always remains finite. At an intruding wedge, like the inner part of a waveguide, all field components remain finite.

*γ*

_{x}and

*γ*

_{y}), we have thus reduced the three-dimensional scattering problem to a two-dimensional numerical problem.

## 5. Numerical implementation

*γ*

_{x}and

*γ*

_{y}, and depending on the size

*z*-direction. For large imaginary

*γ*

_{z}, the mode will only penetrate into the hole or pit a very small distance. It is therefore reasonable to expect that only the modes with a small imaginary

*γ*

_{z}will contribute to the total result. Roberts [17

17. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A **4**, 1970–1983 (1987). [CrossRef]

15. F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. **95**, 103,901 (2005). [CrossRef] [PubMed]

*N*and the second with a smaller number

*Ñ*, we define the following measure:

*V*

_{p}of the hole (or pit). Here, (

**E**

_{Ñ},

**H**

_{Ñ}) is the elec tromagnetic field inside the hole for which the series are truncated after

*Ñ*waveguide modes and (

**E**

_{N},

**H**

_{N}) is the electromagnetic field inside the hole obtained by truncation after

*N*waveguide modes. Hence, this measure corresponds to the error in the energy.

*and*

**α**̄*′ live in pits or holes that are far apart. Moreover, the integral contains the factor 1/*

**α**̄*k*

_{z}that is singular on the circle given by

*k*

^{2}. As stated before, this is the fingerprint of the

*coupled electromagnetic surface wave*. The integrand is still integrable, but a careful implementation is required.

*z*= +

*D*/2, it does not depend on the following important parameters: the thickness

*D*of the conducting layer; the index of refraction

*n*

_{p}inside the pit or hole; whether the scatterer is a pit or a hole and, in case of a pit, its depth

*d*

_{p}. Consequently, once the interaction integrals are calculated for a certain setup, we can vary these parameters with negligible computational effort. This is a great advantage of our method. The possibility to construct a library of calculated interaction integrals is also beneficial.

## 6. Extraordinary transmission

*L*

_{x}=

*L*

_{y}=

*L*) and the index of refraction above and below the layer as well as inside the pits and holes is taken to be unity.

*z*=

*z*

_{0}is calculated directly from the coefficients of the waveguide modes in the following way:

*α*

_{4}denotes the direction of propagation of the waveguide mode. Note that two waveguide modes that have an opposite direction of propagation but that are otherwise identical together produce a non-zero energy flux.

### 6.1. Extraordinary transmission through a single hole

*π*/

*γ*

_{z}) of the propagating mode, indicating that the interference of this mode with its own reflection is responsible for the increased and decreased transmission. It follows from Fig. 4 that if the lowest order mode is just above cut-off, extraordinary transmission of a factor of about 1.5 seems possible. If the size of the hole is increased further, more and more modes are propagating and the normalized energy flux decreases below unity. Going from

*L*=

*λ*to

*L*= 2

*λ*, the energy flux increases to just below unity. For large holes, one expects an energy flux of unity, of course. The energy flux that is shown, is calculated directly from the coefficients found for the waveguide modes and hence, it is not necessarily the energy that will travel along the

*z*-axis and, possibly, arrive at a far field detector. However, because of the perfect conductor assumption, none of this energy is absorbed.

*λ*/4. Shown is the Poynting vector in the radial direction along a half circle with radius of one wavelength, hence the scattering in the near field. The scattering in the plane in which the incident electric field is polarized can be non-zero along the interface, because the corresponding electric field then points in the

*z*-direction. The scattering in the perpendicular plane can not be along the

*z*-direction, for the tangential field at a perfect conductor must be zero. For a pit with size

*L*=

*λ*/5 the scattering is like that of a dipole, whereas for larger pits the scattering is mainly along the optical axis (the

*z*-axis).

### 6.2. Extraordinary transmission through multiple holes

*z*-axis. The electric field of the incident plane wave is either directed perpendicular to this plane or else parallel to this plane. Fig. 6(a) shows the normalized energy flux through one of two holes and through one of three holes for perpendicular polarization. A modulation of the energy flux as a function of distance between the centers of the holes only occurs for holes that are less than two wavelengths apart. We believe that enhanced or decreased transmission in this case is caused by the coupling of evanescent fields scattered from one hole to the other and/or by polarization rotation at the corners of the hole. The solid lines in Fig. 6(b) show the same calculations, but now for parallel polarization. The modulation of the energy flux is now also present for large distances between the holes. Its period is equal to the wavelength of the incident field. Its amplitude for three holes is twice the amplitude for two holes. Furthermore, the amplitude is proportional to the inverse of the distance between the centers of the holes, as expected for a cylindrical wave. If the propagation direction of the incident plane wave is slightly tilted (dashed and dotted line in Fig. 6(b)) then a phase shift occurs that is equal to the delay that the incident field experiences in reaching the farthest hole as compared to the nearest hole. Fig. 6(c) shows, for the incident field polarized parallel, the normalized energy flux through a hole in the presence of a pit. This pit has its open end at the upper side (dashed, black line) or at the lower side (solid, gray line). The field is incident from above. The modulation period for the latter is half that of the first. Furthermore, the amplitude for both cases is much smaller than for the case of two holes (solid, black line).

*k*

_{z}= 0 that are scattered along the metal surface. These scattered waves cause a periodicity of a wavelength when two scatterers at the same side of the metal layer contribute and a periodicity of half the wavelength when there is only one source, as is the case when one hole is accompanied by a pit with its open end at the non-illuminated side. In this case, the surface wave that is excited at the exit of the hole travels to the pit. It there excites another surface wave that travels back to the hole and interferes. This results in a phase shift that corresponds to twice the distance between the hole and the pit. See also Ref. [9

09. H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. **94**, 053,901 (2005). [CrossRef] [PubMed]

*surface plasmon*(or

*surface plasmon polariton*) is confusing. The ending -

*on*suggests a sort of localization. For a surface plasmon, as described by, for example, Raether [25], this means that it is bound to the interface between the metal and the dielectric. The derivation by Raether of the surface plasmon wave vector (which is only valid for a flat interface) is also valid for an interface between a perfect conductor and a dielectric. Then, the plasmon wave vector is just the wave vector of the light. The penetration depth inside the metal is zero and the charges inside the metal oscillate only in the plane of the interface between the metal and the dielectric. The plasmon - or, better,

*coupled electromagnetic surface wave*- then has a constant field strength in the half space above the metal and is not bound to the surface. The existence and the physical nature of the phenomenon, however, are the same for both a conductor with finite conductivity and an idealized perfect conductor. The only difference is the finite decay length and absorption caused by finite conductivity.

*λ*/20 below the metal layer. The metal layer contains two square holes. The small arrows in Fig. 6(b) indicate the data points for which we calculated the near field as shown in the upper and lower figure. Hence, the incident electric field is polarized in the

*x*-direction. This means that the wave along the surface is mainly propagating energy in the

*x*-direction. For the two setups, the difference is indeed largest for the

*x*-component of the Poynting vector. Note that, near the holes, especially in the upper figure, the

*z*-component of the Poynting vector points towards the metal layer.

## 7. Conclusion

*coupled electromagnetic surface wave*would be more appropriate, as the boundedness of this kind of wave depends on the conductivity of the metal.

## A. The waveguide modes

*z*:

*x̄*

_{p}≡

*x*-

*ȳ*

_{p}=

*y*-

*x̄*

_{p},

*ȳ*

_{p}) is a rectangle function that indicates that the mode functions are identical to zero outside the cross-sectional area of the

*p*-th hole:

*x*) is the Heaviside step function. By using (31) it is easy to see that two different modes are orthogonal in the sense:

*ν**} is an orthonormal system with respect to the scalar product (34).*

_{α¯}*z*-coordinates of the upper and lower end of the pit or hole, respectively. Hence, for holes we have

*D*/2 and

*D*/2, for pits we either have

*D*/2 and

*z*

_{p}

_{2}=

*D*/2 -

*d*

_{p}or we have

*z*

_{p}

_{1}= -

*D*/2 +

*d*

_{p}and

*D*/2. We have introduced the constant factors like exp(

*iγz*

*γ*

_{z}, the moduli of the exponents are always equal to or smaller than unity. In Eq. (35) and (36) we use the sine and cosine functions instead of the exponential function if the propagation constant

*γ*

_{z}is very small as compared to

*k*

_{p}, with smallness parameter ϵ [26]. If we would not do this, for

*γ*

_{z}= 0 for some mode, we would miss the mode function that is linear in

*z*and our set of modes would not be complete [27

27.
The reader might wonder why we do not use the sine and cosine form always instead of using the exponential form only for |*γ*_{z}
/*k*_{p}
| < *ϵ*. However, for large and purely imaginary *γ*_{z}
, the functions cos(*γ*_{z}*z*) and sin(*γ*_{z}*z*) increase exponentially with |*z*|, which is not very convenient for numerical implementation.

*ϵ*= 10

^{-5}. For the

*z*-dependent parts we now have:

*z*-direction if Γ

**<**

*α**k*

_{p}, while for Γ

_{α}>

*k*

_{p}the modes are evanescent. For a square pit or hole (

*L*

_{x}=

*L*

_{y}) with

*L*

_{x}<

*λ*

_{p}/2 all modes are evanescent. Finally, the (rotated) transverse components of the modes are then given by:

*ν*_{α ̄}, which is the part of the transverse field that does not depend on

*z*. This means that the above

*z*-dependent part is only defined up to a constant. We have chosen this constant such that for both TE and TM polarization the waveguide modes have the same order of magnitude.

## B. The plane waves above and below the layer

**E**

_{β},

**H**

_{β}are divided into a part that depends on

*x*and

*y*and a part that depends on

*z*:

*z*:

*z*-dependent part we have the following auxiliary function:

*z*-dependent parts:

*k*,

*k*

_{z}and

*ϵ*in the above equations are either

*k*

_{u},

*ϵ*

_{u}or

*k*

_{l},

*ϵ*

_{l}, depending on

*z*. The

*z*-components of the plane waves are given by:

*k*

_{z}= 0. For the waveguide modes in the previous section, we made sure that, when it happens that

*γ*

_{z}= 0, the set of mode functions is still complete. The plane waves in the upper and lower half spaces, however, form a continuous set, parametrized by -∞ <

*k*

_{x},

*k*

_{y}< ∞. The plane waves with

*k*

_{z}= 0 are only a set of measure zero in the space of all plane waves and are therefore irrelevant for the completeness.

## C. The interaction integral

*, via the scattered field through operator A , with another waveguide mode*

**α**̄*′. We will first write out the integral and then we will discuss the numerical implementation.*

**α**̄### C.1. The interaction integral

*β*

_{2}= (

*k*

_{x},

*k*

_{y}) and that ∫d

*β*

_{2}= ∫∫d

*k*

_{x}d

*k*

_{y}. We consider one of the scalar products, with

*= (*

**β***β*

_{1},

*β*

_{2}) and

*β*

_{1}= S,P:

*j*=

*x*,

*y*are given by:

*F*

_{α ̄}

^{β1}(

*k*

_{x},

*k*

_{y}) is zero for all (

*k*

_{x},

*k*

_{y}) when the waveguide mode is TM polarized and the plane wave is S-polarized. Hence, these polarizations do not interact. For the interaction integral we now have:

_{x}=

*x*

^{p}′

_{0}-

_{y}=

*y*

^{p}

*′*

_{0}-

*p*′ and

*p*. From the above equation it is clear that this double integral is difficult for two reasons. First, the factor exp[

*i*(

*k*

_{x}Δ

_{x}+

*k*

_{y}Δ

_{y})] oscillates violently when pit

*P*′ and

*p*are far apart. Second, the factor

*k*

^{2}. The integrand is still integrable, but care has to be taken.

### C.2. Numerical computation of the interaction integrals

*k*

_{x},

*k*

_{y})-plane where the square root

*k*

_{z}= (

*k*

^{2}-

^{1/2}is equal to zero and the exponential factor exp(

*ik*

_{x}Δ

_{x}+

*ik*

_{y}Δ

_{y}) that oscillates violently when the two holes or pits under consideration are far apart. The square root term would be best tackled with polar coordinates, whereas the exponential term would be easier to integrate with cartesian coordinates. To overcome this problem, we split the integration area in 12 domains. See Fig. 9. From symmetry properties of the integrand, it follows that it suffices to integrate over half the (

*k*

_{x},

*k*

_{y})-plane [28]. Furthermore, in the domains 1, 6 and 11 that are situated within the circle

*k*

^{2}, only the real parts need to be calculated. For the other domains, we only need the imaginary parts:

*I*

_{α ̄α ̄′}, is shorthand notation for the integrand of the interaction integral.

*k*-

*δ*)

^{2}≤

*k*

^{2}+

*k*

^{2}≤

*k*

^{2}] and the outer d-ring [(

*k*

^{2}<

*k*+

*δ*)

^{2}]. The value of

*δ*is chosen such, that the number of oscillations of the exponential factor inside the rings is small. However, the d-rings should be wide enough to contain the steepest part of the square root factor (typically larger than or equal to

*k*/10). Within the two d-rings, we choose a polar coordinate system. Furthermore, we apply a substitution to get rid of the square root singularity. We then use a standard, adaptive quadrature routine from the NAG foundation toolbox for Matlab, D01FCF [29

29. This routine is a multi-dimensional adaptive quadrature over a hyper-rectangle. For a description, see the internet link: http://www.nag.com/nagware/mt/doc/d01fcf.html.

*K*

_{1}and

*k*

_{max}. Here, we approximate the slowly varying part parabolically and we integrate exactly this parabolic approximation times the exponent.

## Acknowledgements

## References and links

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

02. | J. Meixner and W. Andrejewski, “Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vol-lkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm,” Annalen der Physik |

03. | C. Flammer, “The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions,” J. Appl. Phys. |

04. | C. Flammer, “The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems,” J. Appl. Phys. |

05. | C.J. Bouwkamp, “Diffraction theory,” Reports on progress in physics |

06. | T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

07. | H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a subwavelength slit: Waveg-uiding and optical vortices,” Phys. Rev. E |

08. | J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, “Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit,” Phys. Rev. E |

09. | H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. |

10. | M.G. Moharam and T.K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A |

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

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

13. | L. Martin-Moreno and F.J. Garcia-Vidal, “Optical transmission through circular hole arrays in optically thick metal films,” Opt. Express |

14. | S.H. Chang, S.K. Gray, and G.C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express |

15. | F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. |

16. | F.J. Garcia de Abajo, “Light transmission through a single cylindrical hole in a metallic film,” Opt. Express |

17. | A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A |

18. | J.M. Brok and H.P Urbach, “A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording,” J. Mod. Opt. |

19. | J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

20. | A.P Hibbins, B.R. Evans, and J.R. Sambles, “Experimental verification of designer surface plasmons,” Science |

21. |
Here and henceforth the square root of a complex number |

22. | J.D. Jackson, |

23. |
Although x,y) is real, we include the conjugation for consistency of the notation. |

24. | J. van Bladel, |

25. | H. Raether, |

26. |
Although the ± for |

27. |
The reader might wonder why we do not use the sine and cosine form always instead of using the exponential form only for | |

28. | The interaction integral also has some other properties that can save a lot of computation time. These properties are outside the scope of this paper. |

29. | This routine is a multi-dimensional adaptive quadrature over a hyper-rectangle. For a description, see the internet link: http://www.nag.com/nagware/mt/doc/d01fcf.html. |

**OCIS Codes**

(050.1220) Diffraction and gratings : Apertures

(050.1960) Diffraction and gratings : Diffraction theory

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 27, 2006

Revised Manuscript: March 16, 2006

Manuscript Accepted: March 16, 2006

Published: April 3, 2006

**Citation**

J. M. Brok and H. P. Urbach, "Extraordinary transmission through 1, 2 and 3 holes in a perfect conductor,
modelled by a mode expansion technique," Opt. Express **14**, 2552-2572 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2552

Sort: Year | Journal | Reset

### References

- H.A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163 (1944). [CrossRef]
- J. Meixner and W. Andrejewski, "Strenge Theorie der Beugung ebener elektromagnetischer Wellen and der vollkommen leitenden Kreisscheibe und an der kreisformigen Offnung im vollkommen leitenden ebenen Schirm," Annalen der Physik 7, 157-168 (1950). [CrossRef]
- C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. I. Oblate spheroidal vector wave functions," J. Appl. Phys. 24, 1218-1223 (1953). [CrossRef]
- C. Flammer, "The vector wave function solution of the diffraction of electromagnetic waves by circular disks and apertures. II. The diffraction problems," J. Appl. Phys. 24, 1224-1231 (1953). [CrossRef]
- C.J. Bouwkamp, "Diffraction theory," Reports on progress in physics 17, 35-100 (1954). [CrossRef]
- 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, 667-669 (1998). [CrossRef]
- H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, "Light transmission through a subwavelength slit: Waveguiding and optical vortices," Phys. Rev. E 67, 036,608 (2003). [CrossRef]
- J. Bravo-Abad, L. Martin-Moreno, and F.J. Garcia-Vidal, "Transmission properties of a single metallic slit: From the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 026,601 (2004). [CrossRef]
- H.F. Schouten, N. Kuzmin, G. Dubois, T.D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E.R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053,901 (2005). [CrossRef] [PubMed]
- M.G. Moharam and T.K. Gaylord, "Rigorous coupled-wave analysis of metallic surface-relief gratings," J. Opt. Soc. Am. A 3, 1780-1787 (1986). [CrossRef]
- J.A. Porto, F.J. Garcia-Vidal, and J.B. Pendry, "Transmission resonances on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 2845-2848 (1999). [CrossRef]
- Q. Cao and P. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057,403 (2002). [CrossRef] [PubMed]
- L. Martin-Moreno and F.J. Garcia-Vidal, "Optical transmission through circular hole arrays in optically thick metal films," Opt. Express 12, 3619-3628 (2004). [CrossRef]
- S.H. Chang, S.K. Gray, and G.C. Schatz, "Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films," Opt. Express 13, 3150-3165 (2005). [CrossRef] [PubMed]
- F.J. Garcia-Vidal, E. Moreno, J.A. Porto, and L. Martin-Moreno, "Transmission of light through a single rectangular hole," Phys. Rev. Lett. 95, 103,901 (2005). [CrossRef] [PubMed]
- F.J.G.I. de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475-1484 (2002). [PubMed]
- A. Roberts, "Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen," J. Opt. Soc. Am. A 4, 1970-1983 (1987). [CrossRef]
- J.M. Brok and H.P. Urbach, "A mode expansion technique for rigorously calculating the scattering from 3D subwavelength structures in optical recording," J. Mod. Opt. 51, 2059-2077 (2004). [CrossRef]
- J.B. Pendry, L. Martin-Moreno, and F.J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847 (2004). [CrossRef] [PubMed]
- A.P. Hibbins, B.R. Evans, and J.R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670 (2005). [CrossRef] [PubMed]
- Here and henceforth the square root of a complex number z is defined such that for real z > 0 we have √z > 0 and √z = +i√z, with the branch cut along the negative real axis.
- J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
- Although υ¯α(x,y) is real, we include the conjugation for consistency of the notation.
- J. van Bladel, Singular Electromagnetic Fields and Sources, 1st ed. (Clarendon Press, Oxford, 1991).
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, 1st ed. (Springer-Verlag, Berlin, 1988).
- <other>. Although the ± for α4 now does not have anything to do with the propagation direction, for consistency, we stick to this notation. </other>
- The reader might wonder why we do not use the sine and cosine form always instead of using the exponential form only for | γz/kp|< ε. However, for large and purely imaginary γz, the functions cos (γzz) and sin (γzz) increase exponentially with |z|, which is not very convenient for numerical implementation.
- The interaction integral also has some other properties that can save a lot of computation time. These properties are outside the scope of this paper.
- This routine is a multi-dimensional adaptive quadrature over a hyper-rectangle. For a description, see the internet link: http://www.nag.com/nagware/mt/doc/d01fcf.html.

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