Abstract

Using the Bloch modes of a periodic, semi-infinite array of slits in a metallic host, we study the transmission of obliquely incident plane-waves through sub-wavelength slits. Matching the tangential E- and H-fields at the entrance facet of the periodic structure yields the complex amplitudes of the various Bloch modes, which exist and propagate within the slit array independently of each other. The computational scheme is robust, convergence is rapid, and a good match at the boundaries is obtained in every case. The regions examined in some detail include the vicinity of the Wood anomaly (where new diffraction orders appear/disappear on the horizon), the neighborhood of a point where surface plasmon polaritons (SPPs) are excited, and an ordinary situation in which the incidence angle is far from the angles that invoke Wood’s anomaly or cause the excitation of SPPs. Field distributions and energy flow diagrams in and around the slits reveal the existence of transmission minima (and reflection maxima) at incidence angles associated with the excitation of SPPs.

© 2006 Optical Society of America

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References

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  1. 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]
  2. R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Proc. Phys. Soc. London 18, 269-275 (1902).
    [CrossRef]
  3. R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
    [CrossRef]
  4. Lord Rayleigh, "On the dynamic theory of gratings," Proc. R. Soc. A 79, 399-416 (1907).
    [CrossRef]
  5. J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
    [CrossRef]
  6. Q. Cao and Ph. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403(4) (2002).
    [CrossRef] [PubMed]
  7. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, "Transmission of light through a periodic array of slits in a thick metallic film," Opt. Express 13, 4485 (2005).
    [CrossRef] [PubMed]
  8. H. Raether, Surface Plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, Berlin, 1986).
  9. J. D. Jackson, Classical Electrodynamics, 3rd edition (Wiley, New York, 1999) Chap. 8.
  10. P. Edward, Handbook of optical constants of solids, 1st edition (Academic Press, 1997).
  11. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient imple-mentation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-76 (1995).
    [CrossRef]
  12. Ph. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]

2006

2005

2002

Q. Cao and Ph. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403(4) (2002).
[CrossRef] [PubMed]

1999

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
[CrossRef]

1996

1995

1935

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
[CrossRef]

1907

Lord Rayleigh, "On the dynamic theory of gratings," Proc. R. Soc. A 79, 399-416 (1907).
[CrossRef]

1902

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Proc. Phys. Soc. London 18, 269-275 (1902).
[CrossRef]

Cao, Q.

Q. Cao and Ph. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403(4) (2002).
[CrossRef] [PubMed]

García-Vidal, F. J.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Lalanne, Ph.

Q. Cao and Ph. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403(4) (2002).
[CrossRef] [PubMed]

Ph. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

Mansuripur, M.

Moharam, M. G.

Moloney, J. V.

Moloney, J. V.

Morris, G. M.

Pendry, J. B.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
[CrossRef]

Pommet, D. A.

Porto, J. A.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
[CrossRef]

Wood, R. W.

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Proc. Phys. Soc. London 18, 269-275 (1902).
[CrossRef]

Wood, R. W.

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
[CrossRef]

Xie, Y.

Zakharian, A. R.

Zakharian, A. R.

J. Opt. Soc. Am. A

Opt. Express

Phys. Rev.

R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
[CrossRef]

Phys. Rev. Lett.

J. A. Porto, F. J. García-Vidal, and J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
[CrossRef]

Q. Cao and Ph. Lalanne, "Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits," Phys. Rev. Lett. 88, 057403(4) (2002).
[CrossRef] [PubMed]

Proc. Phys. Soc. London

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Proc. Phys. Soc. London 18, 269-275 (1902).
[CrossRef]

Proc. R. Soc. A

Lord Rayleigh, "On the dynamic theory of gratings," Proc. R. Soc. A 79, 399-416 (1907).
[CrossRef]

Other

H. Raether, Surface Plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, Berlin, 1986).

J. D. Jackson, Classical Electrodynamics, 3rd edition (Wiley, New York, 1999) Chap. 8.

P. Edward, Handbook of optical constants of solids, 1st edition (Academic Press, 1997).

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Figures (8)

Fig. 1.
Fig. 1.

A TM-polarized plane-wave (Hx , Ey , Ez ) is incident from the air on a semi-infinite slit array at the angle θ. The plane of incidence is yz, the vacuum wavelength of the light is λ o = 1.0 μm, and the slits are in a silver host having εm = - 48.8 + 2.99i. Throughout this paper, the period of the structure is fixed at p = 1.2 μm, and the slit apertures, filled with air, are assumed to have a width w = 0.1μm (i.e., one-tenth of one wavelength). The bottom of the array is placed at z = to eliminate the influence of the light returning from the bottom facet on reflected and transmitted light at the top (entrance) facet of the array.

Fig. 2.
Fig. 2.

Propagation constants of modes numbered 3 to 40 for the slit array depicted in Fig. 1 at θ= 30°; the arrows indicate the direction of increasing mode number. For each mode, the propagation constant along the z-axis, σz , is the same in the slit and in the metallic cladding. Along the y-axis, the propagation constants in the slit are ±σys , while those in the metallic region are ±σym ; see Eq. (1). Parameters of the first two modes (not shown because they are off the charts) are: σz(1) = 1.2119 + 0.0066i, σys(1) = 0.0116 - 0.6847i, σym(1) = 0.2219 + 7.0941i and σz(2) = 0.0002 + 4.8308i, σys(2) = 4.9332 - 0.0002i, σym(2) = 0.3126 + 5.0567i.

Fig. 3.
Fig. 3.

Profiles of the first twelve modes of the slit array depicted in Fig. 1 at θ = 30°; left: Hx , right: Ey . In each case, the field’s magnitude is shown at the top and the corresponding phase profile at the bottom. For display purposes each profile is individually normalized.

Fig. 4.
Fig. 4.

Profiles of Hx and Ey magnitudes across a full period of the slit array in silver host (λ o = 1.0μm, θ= 30°, p = 1.2 μm, w = 0.1 μm). A total of N = 120 modes (in each space) was used to reduce the mismatch between the tangential E and H fields at the interface.

Fig. 5.
Fig. 5.

(a) Convergence of the coefficients of the first five modes of the slit array to their final values, displayed as function of the number of modes, N, used to minimize the mismatch across the z = 0 interface in the case of θ = 30°. (b) Magnitudes of the first 20 modes (obtained with a total of N = 120 modes used to reduce the mismatch) for three different angles of incidence: (blue) θ = 30°, an ordinary case; (green) θ= 9.6°, Wood’s anomaly; (red) θ= 10.2°, SPP excitation. In all cases the host material is silver, λ o = 1.0 μm, p = 1.2 μm, and w = 0.1 μm.

Fig. 6.
Fig. 6.

Reflectivity R (blue), total transmissivity T (red), and transmission efficiency T 1 of the guided mode (green), versus the incidence angle θ for the semi-infinite slit array depicted in Fig. 1 (silver host, p) = 1.2 μm, w = 0.1 μm, λ o = 1.0 μm).

Fig. 7.
Fig. 7.

Field magnitudes (Hx , Ey , Dz ) and Poynting vector S in the yz cross-sectional plane of the array of Fig. 1 (silver host, p = 1.2 μm, w = 0.1 μm, λ o = 1.0 μm) for three different cases. Top row: ordinary behavior at θ= 30°; middle row: Wood’s anomaly at θ= 9.6°; bottom row: SPP anomaly at θ= 10.2°. The color in the Poynting vector diagrams (right-hand column) encodes the magnitude of S , while the arrows indicate the direction of flow of energy.

Fig. 8.
Fig. 8.

Profiles of |Hx |, |Ey |, |Dz | and the Poynting vector S in the yz cross-sectional plane of the array of Fig. 1 at θ= 82.5° (silver host, p = 1.2 μm, w = 0.1 μm, λ o = 1.0 μm). In the Poynting vector plot the color encodes | S |, while the arrows show the direction of flow of energy.

Equations (3)

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H xT n ( y , z ) = { exp ( i k 0 σ z n z ) { h 1 s n exp [ i k 0 σ ys n ( y + w 2 ) ] + h 2 s n exp [ i k 0 σ ys n ( y w 2 ) ] } , w 2 < y < w 2 exp ( i k 0 σ z n z ) { h 1 m n exp [ i k 0 σ ym n ( y w 2 + p ) ] + h 2 m n exp [ i k 0 σ ym n ( y + w 2 ) ] } , w 2 p y w 2
sin ( θ Wood ) + m ( λ o p ) = ± 1 ,
sin ( θ spp ) + m ( λ o p ) = ± Re [ n spp ] .

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