Abstract

Transmission of polarized light through sub-wavelength slit apertures is studied based on the electromagnetic field distributions obtained in computer simulations. The results show the existence of a cutoff for E and a strong transmission (with no cutoff) for E ; here ‖ and ⊥ refer to the direction of the incident E-field relative to the long axis of the slit. These observations are explained by the standard waveguide theory involving inhomogeneous plane waves that bounce back and forth between the interior walls of the slit aperture. We examine the roles played by the slit-width, by the film thickness, and by the real and imaginary parts of the host material’s dielectric constant in determining the transmission efficiency. We also show that the slit’s sharp edges can be rounded to eliminate highly-localized electric dipoles without significantly affecting the slit’s throughput. Finally, interference among the surface charges and currents induced in the vicinity of two adjacent slits is shown to result in enhanced transmission through both slits when the slits are separated by about one half of one wavelength.

© 2004 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, et al, ???Extraordinary optical transmission through subwavelength hole arrays,??? Nature 39, 667-669 (1998).
    [CrossRef]
  2. Ph. Lalanne, J. P. Hugonin, et al, ???One-mode model and Airy-like formulae for one-dimensional metallic gratings,??? J. Opt. Soc. Am. A 2, 48-51 (2000)
  3. J. A. Porto, F. J. Garcia-Vidal, et al, ???Transmission resonances on metallic gratings with very narrow slits,??? Phys. Rev. Lett. 83, 2845-2848 (1999)
    [CrossRef]
  4. N. Bonod, S. Enoch, et al, ???Resonant optical transmission through thin metallic films with and without holes,??? Opt. Express. 11, 482-490 (March 2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-482">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-482</a>
    [CrossRef] [PubMed]
  5. L. Martin-Moreno, F. J. Garcia-Vidal, et al, ???Theory of extraordinary optical transmission through subwavelength hole arrays,??? Phys. Rev. Lett. 86, 1114-1117 (February 2001)
    [CrossRef] [PubMed]
  6. F. Yang and J. R. Sambles, ???Resonant transmission of microwave through a narrow metallic slit,??? Phys. Rev. Lett. 89, 63901(3) (2002)
    [CrossRef]
  7. J. Bravo-Abad, L. Martin-Moreno, et al, ???Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit,??? Phys. Rev. E 69, 26601(69) (2004)
    [CrossRef]
  8. A. R. Zakharian, M. Mansuripur, and J. V. Moloney, ???Transmission of light through small elliptical apertures,??? Opt. Express. 12, 2631-2648 (June 2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2631">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2631</a>
    [CrossRef] [PubMed]
  9. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd edition, Artech House, Boston, 2000
  10. J. D. Jackson, Classical Electrodynamics, Chapter 5, 3rd edition, Wiley, New York, 1999.
  11. D. M. Pozar, Microwave Engineering, Chapter 1, 3rd edition, Wiley, New York, 2005
  12. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, Chapter 7, Wiley, New York, 1991
    [CrossRef]
  13. I. R. Hooper and J.R. Sambles, ???Surface plasmon polaritons on thin metal gratings,??? Phys. Rev. B 67, 235404(7) (2003)
    [CrossRef]
  14. 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]
  15. 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]

J. Opt. Soc. Am. A (1)

Ph. Lalanne, J. P. Hugonin, et al, ???One-mode model and Airy-like formulae for one-dimensional metallic gratings,??? J. Opt. Soc. Am. A 2, 48-51 (2000)

Nature (1)

T. W. Ebbesen, H. J. Lezec, et al, ???Extraordinary optical transmission through subwavelength hole arrays,??? Nature 39, 667-669 (1998).
[CrossRef]

Opt. Express (1)

Opt. Express. (1)

A. R. Zakharian, M. Mansuripur, and J. V. Moloney, ???Transmission of light through small elliptical apertures,??? Opt. Express. 12, 2631-2648 (June 2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2631">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2631</a>
[CrossRef] [PubMed]

Phys. Rev. B (2)

I. R. Hooper and J.R. Sambles, ???Surface plasmon polaritons on thin metal gratings,??? Phys. Rev. B 67, 235404(7) (2003)
[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]

Phys. Rev. E (1)

J. Bravo-Abad, L. Martin-Moreno, et al, ???Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit,??? Phys. Rev. E 69, 26601(69) (2004)
[CrossRef]

Phys. Rev. Lett. (4)

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]

L. Martin-Moreno, F. J. Garcia-Vidal, et al, ???Theory of extraordinary optical transmission through subwavelength hole arrays,??? Phys. Rev. Lett. 86, 1114-1117 (February 2001)
[CrossRef] [PubMed]

F. Yang and J. R. Sambles, ???Resonant transmission of microwave through a narrow metallic slit,??? Phys. Rev. Lett. 89, 63901(3) (2002)
[CrossRef]

J. A. Porto, F. J. Garcia-Vidal, et al, ???Transmission resonances on metallic gratings with very narrow slits,??? Phys. Rev. Lett. 83, 2845-2848 (1999)
[CrossRef]

Other (4)

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd edition, Artech House, Boston, 2000

J. D. Jackson, Classical Electrodynamics, Chapter 5, 3rd edition, Wiley, New York, 1999.

D. M. Pozar, Microwave Engineering, Chapter 1, 3rd edition, Wiley, New York, 2005

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, Chapter 7, Wiley, New York, 1991
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Slit aperture of width W in a metallic film of thickness t. The material of the film (silver) has refractive index n+iκ=0.23+6.99i and permittivity ε=(n+iκ)2=-48.8+3.16i at λo=1.0 µm. The incident beam is uniform along x, and has a broad Gaussian profile along the y-axis; its linear polarization state, denoted by E or E , indicates the incident E-field’s direction relative to the slit’s long axis. The relevant E- and H-field components for the two polarization states are shown below the slit. In our FDTD simulations the perfectly matched layer (PML) boundary condition was applied at the grid boundaries [9].

Fig. 2.
Fig. 2.

Computed plots of Ex, Hy and Hz for the case of E illumination: (top) magnitude, (bottom) phase. Film thickness t=800nm; slit-width W=400nm<½λo. The E-field drops into the aperture, with its magnitude decaying rapidly in the propagation direction -z. Since the E-field has no component perpendicular to the various metallic surfaces, no surface charges appear in this system. The H-field lines bend into the aperture, then turn around and return to the incidence space. The surface currents that support the H-field in the vicinity of the various surfaces are everywhere in the x-direction.

Fig. 3.
Fig. 3.

Computed plots of Ey, Ez, Hx for E illumination: (top) magnitude, (bottom) phase. t=700 nm; Wo/10=100nm. The slit supports a guided mode, whose partial reflection at the bottom of the aperture is responsible for the observed interference fringes in Ey and Hx through the depth of the slit. The E-field plots show the accumulation of electrical charges at various locations on the metallic surfaces; in particular, Ey and Ez at the four corners of the slit show the presence of a strong dipole at the top, and a slightly weaker dipole at the bottom of the slit; the top and bottom dipoles have nearly the same phase. Additionally, Ey shows a periodically varying surface charge density through the slit’s depth. The surface currents that support the magnetic field Hx near the metallic surfaces travel in the ± y-direction on the top surface and in the ± z-direction along the slit walls. These currents not only sustain the adjacent H-fields, but also (through their non-zero divergence) produce the surface charges.

Fig. 4.
Fig. 4.

Computed plots of Ey, Ez, Hx for the case of t=300nm, W=100nm, under E illumination: (top) magnitude, (bottom) phase. The induced electric dipoles localized around the sharp corners of the slit at the top and bottom have nearly identical strengths. The main difference between this case and that in Fig.3 is that the dipoles at the top and bottom of the slit, having a phase difference of ~180°, are oppositely oriented.

Fig. 5.
Fig. 5.

Computed plots of Ey, Ez, Hx for the case of t=500 nm, W=100 nm, under E illumination: (top) magnitude, (bottom) phase. The main difference between this case and those shown in Figs. 3 and 4 is the much weaker electric dipole at the top of the slit relative to that induced at the bottom. The overall transmission efficiency is only about 40% of that obtained in Figs. 3 and 4.

Fig. 6.
Fig. 6.

Computed plots of the energy flux density Sz at the output aperture (z=-½ t) for different film thicknesses ranging from t=100 nm to 900 nm. For reference, the broad blue line in (a)–(c) represents the incident beam’s Sz at z=0 (in free-space). (a) E illumination, W=100nm; (b) E illumination, W=400nm; (c) E illumination, W=600nm; (d) total transmitted Sz (integrated across the aperture and normalized by the incident flux) versus t for the slits depicted in (a)–(c).

Fig. 7.
Fig. 7.

Transmission efficiency of four different slits versus the thickness t of the silver film under E illumination. For smaller values of the slit-width W the peak efficiency is greater and the transmission is a sharper function of t, even though the total amount of light passing through the slit is a decreasing function of W.

Fig. 8.
Fig. 8.

Computed plots of |Ey | for the four materials listed in Table 1. From left to right: Silver, Materials I, II, III; in all cases W=100 nm. To maximize throughput, the first three films have thickness t=700nm, while for the last one t=750 nm. The white curves beneath each slit show the z-component Sz of the transmitted Poynting vector.

Fig. 9.
Fig. 9.

Computed plots of Sy , the component of S perpendicular to the slit wall, along the depth of the slit under E illumination. For the right-hand wall depicted here Sy is positive. The slit transmission is at a maximum for the chosen parameters (t=700nm, W=100nm).

Fig. 10.
Fig. 10.

Logarithmic plots of the magnitude S of the Poynting vector S in the vicinity of the slit under E illumination. The superposed arrows show the orientation of S . Film thickness t=700nm; slit width W=100nm. From left to right, the assumed material is Silver, Material_I, Material_II.

Fig. 11.
Fig. 11.

Computed plots of |Hx | for different degrees of sharpness of the slit’s edges (E illumination). From left to right, corner radius r=0 (i.e., sharp edge), 30nm, 40nm, and 60nm. The four corners in each frame have identical radii. The white curve beneath each slit is a plot of the Poynting vector component Sz immediately below the aperture.

Fig. 12.
Fig. 12.

Computed plots of Ez in the case of E illumination, showing the interaction between a pair of adjacent slits (W=100nm) in a 700nm-thick silver film. The incident Gaussian beam’s FWHM is 10 µm. Left to right: center-to-center spacing of the slits is d=200nm, 500nm, 900nm. The white curves beneath each slit show Sz at the exit facet.

Fig. 13.
Fig. 13.

Left to right: plots of Ey, Ez, Hx for a pair of adjacent slits (W=100nm) in a 700nm-thick silver film under E illumination: (top) magnitude, (bottom) phase. The incident Gaussian beam’s FWHM is 10 µm, and the center-to-center spacing of the slits is d=600 nm.

Tables (1)

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Table 1. Optical constants of silver and three related (artificial) materials

Equations (11)

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ε ( ω ) = ε + Δ ε 1 + i ε τ i σ ω ε 0 .
R = 1 ε + i ε 1 + ε + i ε 2
E y ( y , z ) = E o exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ]
H y ( y , z ) = σ z ( E o Z o ) exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ]
H z ( y , z ) = σ y ( E o Z o ) exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ]
E x ( y , z ) = E o { exp [ i ( 2 π λ o ) σ y y ] + exp [ i ( 2 π λ o ) σ y y ] } exp [ i ( 2 π λ o ) σ z z ] ,
σ y + ε 1 + σ y 2 σ y ε 1 + σ y 2 = exp [ i 2 π σ y W λ o ] .
H x ( y , z ) = H o exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ]
E y ( y , z ) = σ z ( Z o H o ε ) exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ]
E z ( y , z ) = σ y ( Z o H o ε ) exp [ i ( 2 π λ o ) ( σ y y + σ z z ) ] .
ε σ y + ε 1 + σ y 2 ε σ y ε 1 + σ y 2 = exp [ i 2 π σ y W λ o ] .

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