Abstract

The anomalously-high transmission of light through sub-wavelength apertures is a phenomenon which has been observed in numerous experiments, but whose theoretical explanation is incomplete. In this article we present a numerical analysis of the power flow (characterized by the Poynting vector)of the electromagnetic field near a sub-wavelength sized slit in a thin metal plate, and demonstrate that the enhanced transmission is accompanied by the annihilation of phase singularities in the power flow near the slit.

© 2003 Optical Society of America

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References

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  1. J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
    [Crossref]
  2. J.F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, 1999).
  3. M.S. Soskin and M.V. Vasnetsov, “Singular Optics”, in Progress in Optics42, ed. E. Wolf (Elsevier, Amsterdam, 2001), 219–276.
  4. 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]
  5. T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec, and T.W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
    [Crossref]
  6. A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focusof a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [Crossref]
  7. F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).
  8. G.P. Karman, M.W. Beijersbergen, A. van Duijl, and J.P. Woerdman, “Creation and annihilation of phase singularities in a focal field,” Opt. Lett. 22, 1503–1505 (1997).
    [Crossref]
  9. T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
    [Crossref]
  10. H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.
  11. D.E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972, 3rd edition).

2001 (1)

1999 (1)

T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
[Crossref]

1998 (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 391, 667–669 (1998).
[Crossref]

1997 (1)

1974 (1)

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

1972 (1)

F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).

1967 (1)

Beijersbergen, M.W.

Berry, M.V.

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

Blok, H.

T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.

Boivin, A.

Dow, J.

Ebbesen, T.W.

T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec, and T.W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[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]

Ghaemi, H.F.

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]

Karman, G.P.

Landstorfer, F.

F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).

Lenstra, D.

T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.

Lezec, H.J.

T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec, and T.W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[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]

Linke, R.A.

Meinke, H.

F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).

Niedermair, G.

F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).

Nye, J.F.

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

J.F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, 1999).

Pellerin, K.M.

Schouten, H.F.

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.

Soskin, M.S.

M.S. Soskin and M.V. Vasnetsov, “Singular Optics”, in Progress in Optics42, ed. E. Wolf (Elsevier, Amsterdam, 2001), 219–276.

Thio, T.

T. Thio, K.M. Pellerin, R.A. Linke, H.J. Lezec, and T.W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[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]

van Duijl, A.

Vasnetsov, M.V.

M.S. Soskin and M.V. Vasnetsov, “Singular Optics”, in Progress in Optics42, ed. E. Wolf (Elsevier, Amsterdam, 2001), 219–276.

Visser, T.D.

T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
[Crossref]

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.

Woerdman, J.P.

Wolf, E.

Wolff, P.A.

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]

IEEE J. Quant. Elect. (1)

T.D. Visser, H. Blok, and D. Lenstra, “Theory of polarization-dependent amplification in a slab waveguide with anisotropic gain and losses,” IEEE J. Quant. Elect. 35, 240–249 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

Nachrichtentechn. Z. (1)

F. Landstorfer, H. Meinke, and G. Niedermair, “Ringförmiger Energiewirbel im Nahfeld einer Richtantenne,” Nachrichtentechn. Z. 25, 537–576 (1972).

Nature (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 391, 667–669 (1998).
[Crossref]

Opt. Lett. (2)

Proc. Roy. Soc. Lond. A (1)

J.F. Nye and M.V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

Other (4)

J.F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, 1999).

M.S. Soskin and M.V. Vasnetsov, “Singular Optics”, in Progress in Optics42, ed. E. Wolf (Elsevier, Amsterdam, 2001), 219–276.

H.F. Schouten, T.D. Visser, D. Lenstra, and H. Blok, “Light transmission through a sub-wavelength slit: waveguiding and optical vortices,” Phys. Rev. E, in press.

D.E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972, 3rd edition).

Supplementary Material (2)

» Media 1: GIF (186 KB)     
» Media 2: GIF (212 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Illustrating the relation between phase singularities (a), stationary points of the phase (c,e), and the corresponding singularities of the power flow (b,d,f). The arrows in the left-hand column indicate the direction of increasing phase Φ E .

Fig. 2.
Fig. 2.

Illustrating the notation relating to transmission through a slit.

Fig. 3.
Fig. 3.

Illustration of the power flow in the neighborhood of a 200 nm wide slit in a 100 nm thick plate of evaporated silver, with wavelength λ = 500 nm and n = 0.05+i2.87 (the value of the refractive index wastak en from [11]). Features (a) and (d) are left-handed centers, (b) and (c) are right-handed centers, and (e) and (f) are saddles. For this example the transmission coefficient T = 1.11. The color coding indicatesthe modulus of the (normalized) Poynting vector. The dashed box indicatest he region illustrated in Movie 1.

Fig. 4.
Fig. 4.

(186 KB) The field of power flow asa function of the slit width w in the region indicated in Fig. 3. The four phase singularities move together as the slit width is increased, and finally annihilate, leaving a smoother field of power flow corresponding to a higher transmission coefficient T.

Fig. 5.
Fig. 5.

Array of singularities in the field of power flow in the neighborhood of the slit. All parametersar e as given in Fig. 3. Left-handed vortices(cen ters) and righthanded vortices (centers) are denoted by LV and RV, respectively, and saddles are denoted by S.

Fig. 6.
Fig. 6.

(212 KB) Schematic of the position and type of singularities of power flow near a sub-wavelength slit. It is to be observed that multiple creation and annihilation events occur as the slit width is gradually increased. Left-handed vortices (centers) and right-handed vortices (centers) are denoted by LV and RV, respectively, and saddlesar e denoted by S. U denotesa vortex (center) very close to a saddle point which cannot be spatially resolved by the particular grid used for these calculations; it can be seen that eventually the singularities separate sufficiently to be distinguishable.

Equations (9)

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S ( x , z ) = 1 2 Re { E ̂ ( x , z ) × H ̂ * ( x , z ) } ,
sin ϕ S ( x , z ) S z ( x , z ) S ,
cos ϕ S ( x , z ) S x ( x , z ) S ,
S ( x , z ) = 1 2 ω μ 0 Im { E ̂ y E ̂ y * } .
E ̂ y ( x , z ) = E ̂ y ( x , z ) e i ϕ E ( x , z ) .
S ( x , z ) = 1 2 ω μ 0 E ̂ y ( x , z ) 2 ϕ E ( x , z ) .
s E 1 2 π C ϕ E · d r ,
E ̂ i ( x , z ) = E ̂ i inc ( x , z ) i ω Δ ε D G ̂ ij E ( x , z ; x , z ) E ̂ j ( x , z ) d x d z ,
T slit S z d 2 x + plate ( S z S z inc ) d 2 x slit S z ( 0 ) d 2 x .

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