## Surface plasmon polaritons on metallic surfaces

Optics Express, Vol. 15, Issue 1, pp. 183-197 (2007)

http://dx.doi.org/10.1364/OE.15.000183

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### Abstract

A unified theoretical study of surface plasmon polaritons on flat metallic surfaces and interfaces is undertaken to clarify the nature of these electromagnetic waves, conditions under which they are launched, and the restrictions imposed by Maxwell’s equations that ultimately determine the strength of the excited plasmons. Finite Difference Time Domain computer simulations are used to provide a clear picture of the electromagnetic field distribution and the energy flow profile in a specific situation. The examined case involves the launching of plasmonic waves on the entrance facet of a metallic host perforated by a subwavelength slit, and the (simultaneous) excitation of the slit’s guided mode.

© 2007 Optical Society of America

## 1. Introduction

^{1-4}This flurry of activity has, in turn, reawakened interest in surface plasmon polaritons (SPPs) and inspired theoretical research in this area.

^{5-8}Although the fundamental properties of SPPs have been known for nearly five decades,

^{9,10}there remain certain subtle issues that could benefit from further critical analysis.

^{11}

## 2. General Formulation

*ε*the propagation vector is

*k*=

*k*

_{o}(

*σ*

_{y}*y*̂ +

*σ*

_{z}*z*̂), where

*k*

_{o}= 2π/λ

_{o}and σ

_{y}

^{2}+ σ

_{z}

^{2}=

*ε*. In general,

*σ*is allowed, corresponding to the solution that approaches zero when

_{z}*z*→ ±∞. This is why

*σ*

_{z1}of the upper cladding in Fig. 1 is chosen to have a plus sign, whereas that of the lower cladding has a minus sign. (

*σ*

_{z1},

*σ*

_{z2}have positive imaginary parts.)

*H*

_{o}is the (complex) amplitude of the magnetic field,

*ωt*) shall be omitted from the equations.

*ε*

_{1}. In general, the modal fields are either odd or even with respect to the

*y*-axis, allowing one to express the

*H*-field profile of a given mode as follows (plus sign for even, minus sign for odd modes):

*E*-field for each mode can be found from Eqs. (1). Continuity of

*H*and

_{x}*E*at the

_{y}*w*boundaries yields

*H*

_{1}from Eq. (3a) into Eq. (3b), rearranging the terms, and expressing

*σ*

_{z1}and

*σ*

_{z2}in terms of

*σ*, we find:

_{y}*σ*=

_{y}*σ*

^{(r)}

_{y}+ i

*σ*

^{(i)}

_{y}is the characteristic equation of the waveguide depicted in Fig. 1. Each solution

*σ*of Eq. (4) corresponds to a particular mode of the waveguide; when the plus (minus) sign is used on the right-hand side of Eq.(49), the solution represents an even (odd) mode. Since we are presently interested in modes that propagate from left to right in Fig. 1, the imaginary part of

_{y}*σ*must be non-negative (i.e.,

_{y}*∂*

^{(i)}

_{y}≥ 0), otherwise the mode will grow exponentially as

*y*→ ∞. Also, when computing the complex square roots in Eq. (4), one must always choose the root which has a positive imaginary part.

*r*for a

_{p}*p*-polarized (TM) plane-wave at the interface between media of dielectric constants

*ε*

_{1}and

*ε*

_{2}. The Fresnel coefficient has a singularity (pole) at

*w*→ ∞, the complex exponential approaches zero and the pole itself becomes a solution. This can be seen most readily with reference to Eqs. (3); by allowing exp(+i

*k*

_{o}

*σ*

_{z2}

*w*/2) → 0 and substituting for

*H*

_{1}from Eq. (3a) into Eq. (3b), we find

*σ*

_{z2}/

*ε*

_{2}= -

*σ*

_{z1}/

*ε*

_{1}, namely,

*σ*

_{z2}= 0, which, when plugged into Eqs. (2, 3), reveals the electromagnetic field to be identically zero everywhere.

*w*→ 0, the complex exponential in Eq. (4) approaches unity, and

*σ*

_{z1}= 0, and yields a single plane-wave that propagates along the

*y*-axis throughout the entire space; the plane-wave will be homogeneous if

*ε*

_{1}happens to be real and positive, otherwise it will be inhomogeneous.

*σ*|, we note that

_{y}*σ*happens to be in the second quadrant of the complex plane, that is, it must be written as -(

_{y}*σ*

^{(r)}

_{y}/|

*σ*

^{(r)}

_{y}|)

*σ*

^{(i)}

_{y}+ i|

*σ*

^{(r)}

_{y}|. The simplified form of Eq. (4) in the limit of large |

*σ*| thus becomes:

_{y}*ε*

_{1}and

*ε*

_{2}in the complex plane happens to be greater than 90°. Thus, in the limit of large |

*σ*|, the value of |

_{y}*σ*

^{(r)}

_{y}| is fixed by |(

*ε*

_{1}+

*ε*

_{2})/(

*ε*

_{1}− ε

_{2})|. Subsequently, the choice of an even or odd solution (i.e., ± sign on the right-hand-side of Eq. (5)), the phase of (

*ε*

_{1}+

*ε*

_{2})/(

*ε*

_{1}−

*ε*

_{2}), and a ± sign choice for

*σ*

^{(r)}

_{y}determine the value of

*σ*

^{(i)}

_{y}. An infinite number of such solutions for

*σ*

^{(i)}

_{y}exist, as the left-hand-side of Eq. (5) is repeated whenever

*σ*

^{(i)}

_{y}increases by

*λ*

_{o}/

*w*.

## 3. Metallic slab in free space

*ε*

_{1}= 1.0,

*ε*

_{2}= -19.6224 + 0.443i (silver at

*λ*

_{o}= 650 nm). Fixing the slab’s thickness at

*w*= 50 nm and searching the complex-plane for solutions of Eq. (4) yields the first few values of

*σ*

^{(±)}

_{y}=

*σ*

^{(r)}

_{y}+ iσ

^{(i)}

_{y}listed in Table 1; the ± superscripts identify the even and odd modes, respectively. Only solutions having non-negative values of

*σ*

^{(i)}

_{y}are considered so that, as

*y*→ +∞, the corresponding modes will decay to zero. Although we will be concerned mainly with the top two (fundamental) solutions in Table 1, there exists an infinite number of solutions with large values of

*σ*

^{(i)}

_{y}. The latter are generally needed to match the boundary conditions upon launching a SPP, but, due to their rapid decay along the

*y*-axis, the modes with large

*σ*

^{(i)}

_{y}do not appear to have any practical significance otherwise.

*w*increases, the fundamental solutions (highlighted in Table 1) approach each other, reaching the common value of

*σ*

_{spp}). As

*w*→ 0, the even solution approaches

*σ*

_{y}

^{(r)}and a fairly large

*σ*

_{y}

^{(i)},. Table 2 lists the fundamental solutions of Eq. (4) for a range of values of

*w*.

### 3.1. Polarization dependence of SPP

*cannot*support SPPs at metal-dielectric interfaces. The following argument proves the same point by appealing to the underlying physics of surface plasmons. For the sake of simplicity we consider a thick metal plate in vacuum; see Fig. 2. A SPP consists of two inhomogeneous plane-waves (one in the free space, the other in the metal), both having the same

*σ*in the propagation direction (phase velocity

_{y}*V*=

_{p}*c*/

*σ*). The diagram in Fig. 2(a) represents a true SPP, with the

_{y}*E*-fields originating on positive (surface) charges and terminating on negative ones. If the continuity of

*H*

_{∥}at the surface is assumed, then a negative ε

_{metal}ensures the continuity of

*D*

_{⊥}, and

*E*

_{∥}can be made continuous by the proper choice of

*σ*, namely,

_{y}*σ*=

_{y}*σ*. In contrast, the diagram in Fig. 2(b) presents a physical impossibility: absence of magnetic charges in nature means that the

_{spp}*H*-field must be divergence-free everywhere and, in particular, at the metal-vacuum interface; however, since

*H*

_{∥}will now have opposite directions above and below the surface, it cannot satisfy the requisite boundary condition. This is the reason why SPPs must, of necessity, be TM-polarized.

### 3.2. Dependence of SPP on wavelength: group velocity

*σ*

_{2}(

*ω*) = 1 −

*ω*

_{p}^{2}/(

*ω*

^{2}+i

*γω*). Here

*ω*and

_{p}*γ*are the conduction electrons’ plasma frequency and damping coefficient, respectively. The SPP’s characteristic function is thus written

*λ*

_{o}= 650 nm (corresponding to

*ω*= 0.29 × 10

^{16}rad/s) is

*ε*= -19.6224 + 0.443i. The known value of

*ε*at this frequency then yields

*ω*= 1.317 × 10

_{p}^{16}rad/s and

*γ*= 0.623 × 10

^{14}s

^{-1}.

**Note:**Although the chosen value of

*ε*at

*λ*

_{o}= 650 nm is that of silver, its value at other wavelengths deviates from silver’s, the reason being that the Drude model ignores the contribution of bound electrons to the optical properties of the material. Dionne

*et al*.

^{8}have used the experimentally-determined

*ε*(

*ω*) in their calculations, and pointed out the consequences for SPP’s dispersion relation; for our purposes here, however, the Drude model should suffice.

*ε*(

*ω*) and

*σ*(

_{y}*ω*) in a frequency range extending from the near infrared to the ultraviolet are shown in Fig. 3. The two critical frequencies specifically marked on these plots are

*ω*, where

_{p}*Re*[

*ε*] ≈ 0, and

*Re*[

*ε*] ≈ -1, the latter being a singularity of

*σ*(

_{y}*ω*). Surface plasmon polaritons exist below

*Im*[

*σ*(

_{y}*ω*)] ≈ 0 and

*Re*[

*σ*(

_{y}*ω*)]>1. Between

*ω*, in the so-called SPP bandgap,

_{p}^{8}the large values of

*Im*[

*σ*(

_{y}*ω*)] preclude the possibility of long-range surface waves. Above

*ω*, the dielectric function

_{p}*ε*(

*ω*) is essentially real and positive; the material, therefore, is transparent, with a refractive index

*Re*[

*σ*(

_{y}*ω*)] < 1 represents incidence at Brewster’s angle

*θ*, where

_{B}*y*= 0 the SPP’s time-dependence is

*f*(

*y*= 0,

*t*) =

*∫F*(

*ω*−

*ω*

_{o})exp(-i

*ωt*)d

*ω*, where

*f*(∙) is expressed as the Fourier transform of a narrowband function

*F*(∙) centered at

*ω*

_{o}. At a later point

*y*=

*y*

_{o}, each Fourier component of

*f*(∙) acquires a

*y*-dependent term as follows:

*ωσ*(

_{y}*ω*) in a Taylor series around

*ω*=

*ω*

_{o}and retaining the first few terms yields,

*ω*are evaluated at

*ω*

_{o}. Consequently,

*y*

_{o}, the second exponential term distorts the waveform’s spectrum (albeit insignificantly, if the spectrum is narrow and/or

*y*

_{o}is small), and the last term causes a delay in the arrival of the pulse at

*y*

_{o}. The group velocity is thus obtained from the last term as

*V*(

_{g}*ω*

_{o}) =

*c*/[

*ωσ*

_{y}

^{(r)}(

*ω*)]′, where the derivative is evaluated at

*ω*=

*ω*

_{o}. Figure 4 shows plots of

*V*(

_{g}*ω*) as well as the distortion coefficients of Eq. (9) in the frequency range (a) below

*ω*. The group velocity diminishes as

_{p}## 4. Slit in metallic host

^{12}we investigated transmission of light through subwavelength slits in metallic slabs, and discussed the role of slab thickness and slit width, among other factors. Here we confine our attention to slits in semi-infinitely thick slabs, to avoid complications arising from the guided mode’s multiple reflections at the slit’s entrance and exit facets.

*ε*

_{2}= 1.0) in a silver host at

*λ*

_{o}= 650 nm. For small values of

*σ*

_{y}

^{(i)}the transcendental equation yields only one solution, highlighted in Table 3, which corresponds to an even, low-loss, guided mode along the length of the slit. [The trivial solution

*σ*

_{y}

^{(i)}decay rapidly along the

*y*-axis; unlike the fundamental solution, none of the modes having large σ

_{y}

^{(i)}can be considered “guided.” However, when an external beam arrives at the entrance facet of the slit, such rapidly-decaying modes are needed to match the boundary conditions. In the system depicted in Fig. 5, for example, the Gaussian beam, arriving from the free-space at the semi-infinite slit, excites the guided mode, but it also excites surface modes that, together with the guided mode and the free-space modes reflected toward the source, help to satisfy Maxwell’s boundary conditions in the

*y*= 0 plane. (In a recent publication devoted to periodic slit arrays,

^{13}we presented a complete modal analysis and showed the utility of lossy modes in determining the optical properties of such arrays. A similar analysis for single slits requires not only the discrete modes listed in Table 3, but also the continuum modes to be described in Section 4.3.)

### 4.1. Effect of the slit width

*σ*for TM modes in slits of varying width; the host medium is silver, the slit is empty, and the incident wavelength is

_{y}*λ*

_{o}= 650 nm. For

*w*≪

*λ*

_{o}there exists only one guided mode (which turns out to be even). With an increasing slit-width, both the real and imaginary parts of

*σ*associated with this guided mode decrease. As

_{y}*w*increases, an odd mode appears which, at first, has a fairly large loss factor, but σ

_{y}

^{(i)}continues to decrease until, at

*w*∼

*λ*

_{o}/2, this odd mode becomes guided as well. Further increases in

*w*bring more and more guided modes into the system; at

*w*= 980 nm, for example, there exists a total of four guided modes (two odd and two even).

### 4.2. Launching a guided mode

*w*= 200nm,

*ε*

_{2}= 1.0) in a silver host. A Gaussian beam (

*λ*

_{o}= 650 nm, FWHM = 4.0 μm) is normally incident at the entrance facet of the slit, as in Fig. 5. The guided mode is seen in Fig. 6 to propagate along the

*y*-axis, and a certain amount of light is reflected back toward the source. (The incident beam having been removed, only reflected and transmitted beams appear in these pictures.) The guided mode’s period along the length of the slit in Fig. 6 is 0.58μm, in agreement with λ

_{o}/

*Re*[

*σ*

_{y}

^{(+)}] = 0.65/1.1225 = 0.579μm; see Table 4.

*z*directions is consistent with the lateral propagation of surface plasmons away from the slit. A fit to the

*H*-field profile of the upward-traveling SPP (see Fig. 8) yields

*λ*= 634 nm and |

_{spp}*H*(

_{x}*y*= 0,

*z*)| ∼ exp(-0.00585

*z*), consistent with the presence of a lone SPP on either side of the slit.

*τ*= 20 fs Gaussian pulse. For this simulation the assumed functional form of

*ε*(ω) was the same as that used in subsection 3.2 (i.e., Drude model). The computed group velocity

*V*= 0.925

_{g}*c*is very close to the theoretical value of 0.9217

*c*(see Fig. 4), and the Gaussian profile of the SPP along the

*z*-axis (FWHM = 5.495μm) is consistent with the value of

*V*= 5.53μm.

_{g}τ*w*. For very small

*w*, the SPP on either side of the slit is weak; this is expected, of course, considering that no SPP can be launched in the absence of a slit. As the slit widens, the SPP becomes strong at first, but weakens again and reaches a minimum when

*w*∼

*λ*

_{o}. We expect the SPP strength to eventually stabilize with further increases in

*w*, as optical interference between the two metal blocks forming the slit’s walls should decline as the blocks move apart.

*z*-axis at the slit’s edges, it should not come as a surprise that this mode is exactly the same as that of a thick slab described in Section 3 and listed in Table 2 under

*w*= ∞; after all, the interface between the free-space and a semi-infinite metallic medium can sustain one and only one confined mode. What is perhaps surprising here is that none of the slit modes computed from Eq. (4) – and, for the specific case of

*w*= 100 nm, listed in Table 3 – resemble the front facet SPPs shown, for instance, in Fig. 8 (

*w*= 200 nm in this case). While the short-range/lossy modes of Table 3 (or their counterparts associated with other values of the slit-width

*w*) exhibit long-range and rapid oscillations along the

*z*-axis, their spatial frequencies (

*σ*

_{z}) generally differ from the observed SPP frequency at the front facet. This observation leads one to suspect the existence of additional modes, i.e., modes that lie beyond the solution space of Eq. (4), at the entrance facet of the waveguide depicted in Fig. 5. In the following subsection we investigate the surface modes of such slit waveguides, uncover the existence of a continuum of modes at the front facet, and show consistency with the above FDTD simulations.

### 4.3. Surface modes of the slit waveguide

*z*= ±∞) are physically admissible so long as the

*z*-component of their

*k*-vector is real (i.e.,

*σ*

_{z1}is real-valued). In general, the field profiles are either odd or even around the

*y*-axis, allowing one to express the

*H*-field of a given mode as follows (± for even and odd modes):

*E*-field components may be found from Eqs. (1). Continuity of

*H*and

_{x}*E*at the

_{y}*σ*

_{z2}in terms of

*σ*

_{z1}(remembering that

*σ*

_{y}

^{2}+

*σ*

_{z1}

^{2}=

*ε*

_{1}and

*σ*

_{y}

^{2}+

*σ*

_{z2}

^{2}=

*ε*

_{2}), we arrive at

*H*

_{1}′. In what follows we shall focus our attention on the plasmonic wave excited on the upper half of the

*z*-axis, namely, the SPP running along the positive

*z*-axis (

*f*(

*z*) along with its spatial Fourier spectrum

*F*(

*σ*). Note that

_{z}*F*(

*σ*) contains primarily positive frequencies located in the neighborhood of the SPP frequency. For every positive

_{z}*σ*

_{z1}in Eq. (10), however, there exists a corresponding negative frequency with the same amplitude (albeit multiplied by -1). Therefore, the negative spatial frequencies associated with the

*f*(

*z*) SPP (beneath the metallic surface) have the spectrum

*G*(

*σ*) of Fig. 13(d) and the spatial profile

_{z}*g*(

*z*) of Fig. 13(c). Note that

*g*(

*z*) is essentially zero on the positive

*z*-axis, its main contribution to the plasmonic wave being in the immediate neighborhood of the origin at

*z*= 0, where it overlaps with the initial part of

*f*(

*z*); see Fig. 13(e). The overlap will be even less significant if the SPP happens to have a sharper rising edge at

*z*= 0. The conclusion is that the presence of both positive and negative spatial frequencies within the continuum of surface modes described by Eq. (10) is not incompatible with the existence of unidirectional plasmons at the entrance facet of the slit.

13. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express **14**,6400–6413 (2006). [CrossRef] [PubMed]

## 5. Concluding remarks

*z*-axis, it decays rapidly in the

*y*-direction; nonetheless, its existence and properties are critical factors in determining the strength of the slit’s guided mode that is launched at the same entrance facet but travels subsequently down the slit along the

*y*-axis. It has been an objective of the present paper to identify and characterize all short-range, lossy modes of metallo-dielectric interfaces, in anticipation of the role that such modes necessarily play in any comprehensive analysis of plasmonic devices. We have presented a modal analysis of periodic arrays of slits in a recent publication;

^{13}the extension of this analysis to the case of single slits (with the help of the lossy modes identified in the present paper) will be the subject of a forthcoming paper.

## Acknowledgements

## References

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

2. | R. D. Averitt, S. L. Westcott, and N. J. Halas, “Ultrafast electron dynamics in gold nanoshells,” Phys. Rev. B |

3. | J. J. Mock, S. J. Oldenburg, D. R. Smith, D. A. Schultz, and S. Schultz, “Composite Plasmon Resonant Nanowires,” Nano Letters |

4. | G. Gay, O. Alloschery, B. V. de Lesegno, C. O’Dwyer, J. Weiner, and H. J. Lezec, “The optical response of nano-structured surfaces and the composite diffracted evanescent wave model,” Nature Phys. |

5. | H. F. Ghaemi, T. Thio, and D. E. Grupp, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B |

6. | F. J. Garcia-Vidal and H. J. Lezec, et al, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Phys. Rev. Lett. |

7. | 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 |

8. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B |

9. | H. Raether, |

10. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

11. | A. Taflove and S. C. Hagness, |

12. | Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through slit apertures in metallic films,” Opt. Express |

13. | Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(310.2790) Thin films : Guided waves

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: September 14, 2006

Manuscript Accepted: December 19, 2006

Published: January 8, 2007

**Citation**

Armis R. Zakharian, Jerome V. Moloney, and Masud Mansuripur, "Surface plasmon polaritons on metallic surfaces," Opt. Express **15**, 183-197 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-1-183

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-69 (1998). [CrossRef]
- R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998). [CrossRef]
- J. J. Mock, S. J. Oldenburg, D. R. Smith, D. A. Schultz, and S. Schultz, "Composite Plasmon Resonant Nanowires," Nano Lett. 2,465 -69 (2002). [CrossRef]
- G. Gay, O. Alloschery, B. V. de Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nano-structured surfaces and the composite diffracted evanescent wave model," Nat. Phys. 264, 262 - 67 (2006). [CrossRef]
- H. F. Ghaemi, T. Thio, and D. E. Grupp, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998). [CrossRef]
- F. J. Garcia-Vidal, H. J. Lezec, et al, "Multiple paths to enhance optical transmission through a single subwavelength slit," Phys. Rev. Lett. 90, 213901(4) (2003). [CrossRef]
- 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, 48-51 (2000). [CrossRef]
- J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, "Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model," Phys. Rev. B 72, 075405 (2005). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1986).
- J. J. Burke, G. I. Stegeman, and T. Tamir, "Surface-polariton-like waves guided by thin, lossy metal films," Phys. Rev. B 33, 5186-5201 (1986). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., (Artech House, 2000).
- Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106-6121 (2004). [CrossRef] [PubMed]
- Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts," Opt. Express 14, 6400-6413 (2006). [CrossRef] [PubMed]

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