Abstract

Using Bloch modes to study the extraordinary transmission of light through a periodic array of slits in a metallic host, we discuss the differing roles of surface plasmon polaritons and Wood’s anomalies in the observed behavior of such structures. Under certain circumstances, the first few excited modes appear to play a decisive role in determining the transmission efficiency of the array. Surface plasmon excitations tend to reduce the transmissivity of a semi-infinitely thick slit array, yet, paradoxically, the same reduction can account for enhanced transmission in an array of finite thickness τ, provided that τ is tuned to a Fabry-Perot-like resonance between the entrance and exit facets of the slit array. At the Wood anomaly, power redistribution produces sharp peaks in the diffraction efficiencies of various reflected and transmitted orders of the semi-infinite structure. With skew incidence, the degenerate states split, resulting in two peaks and two valleys, as observed by Wood in his 1902 experiments.

© 2006 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
    [CrossRef]
  2. M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66,195105-11 (2002).
    [CrossRef]
  3. H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
    [CrossRef] [PubMed]
  4. G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
    [CrossRef]
  5. S. H. Chang, S. Gray, and G. 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]
  6. C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
    [CrossRef]
  7. 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]
  8. 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]
  9. J. A. Porto, F. J. García-Vidal, J. B. Pendry, "Transmission resonance on metallic gratings with very narrow slits," Phys. Rev. Lett. 83, 02845(4) (1999).
    [CrossRef]
  10. J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 26601(6) (2004).
    [CrossRef]
  11. Y. Xie, A.R. Zakharian, J. V. Moloney, M. Mansuripur, "Transmission of light through slit apertures in metallic films," Opt. Express 12, 6106 (2004).
    [CrossRef] [PubMed]
  12. 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]
  13. R. W. Wood, "Anomalous diffraction gratings," Phys. Rev. 48, 928-937 (1935).
    [CrossRef]
  14. H. Raether, Surface Plasmons on smooth and rough surfaces and on gratings, (Springer-Verlag, Berlin, 1986).
  15. Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
    [CrossRef]
  16. Ph. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
    [CrossRef]
  17. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-76 (1995).
    [CrossRef]
  18. Ph. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-84 (1996).
    [CrossRef]
  19. J. D. Jackson, Classical Electrodynamics, Chapter 8, 3rd edition, Wiley, New York, 1999.
  20. A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1983.
  21. R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, New York, 1978.
  22. Lord Rayleigh, "On the dynamic theory of gratings", Proc. R. Soc. A 79, 399-416 (1907).
    [CrossRef]
  23. E. Noponen, "Electromagnetic Theory of Diffractive Optics," dissertation, Dept. of Technical Physics, Helsinki University of Technology, Finland (1994).
  24. P. Edward, Handbook of optical constants of solids, 1st edition, Academic press, 1997.

2006 (1)

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

2005 (3)

2004 (2)

2003 (2)

C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
[CrossRef]

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

2002 (1)

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66,195105-11 (2002).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

1996 (1)

1995 (1)

1935 (1)

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

1907 (1)

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

1902 (1)

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]

Alloschery, O.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Chang, S. H.

Chavel, P.

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Gay, G.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Gaylord, T. K.

Genet, C.

C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Grann, E. B.

Gray, S.

Hugonin, J. P.

Ph. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Ph. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-84 (1996).
[CrossRef]

Lezec, H. J.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Lezec, H. J.

Mansuripur, M.

Moharam, M. G.

Moloney, J. V.

Morris, G. M.

O'Dwyer, C.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Pommet, D. A.

Rodier, J. C.

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Rodier, J. C.

Ph. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Sauvan, C.

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Schatz, G.

Thio, T.

H. J. Lezec and T. Thio, "Diffracted evanescent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12, 3629-3651 (2004).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Treacy, M. M. J.

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66,195105-11 (2002).
[CrossRef]

van Exter, M. P.

C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
[CrossRef]

Viaris de Lesegno, B.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Weiner, J.

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Woerdman, J. P.

C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Wood, R. W.

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

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]

Xie, Y.

Zakharian, A. R.

Zakharian, A.R.

J. Opt. Soc. Am. A (2)

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature 39, 667-669 (1998).
[CrossRef]

Nature Phys. (1)

G. Gay, O. Alloschery, B. Viaris de Lesegno, C. O'Dwyer, J. Weiner, H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 264, 262 - 67 (2006).
[CrossRef]

Opt. Communications (1)

C. Genet, M. P.  van Exter, and J. P. Woerdman, "Fano-type interpretation of red shifts and red tails in hole array transmission spectra," Opt. Communications 225, 331 (2003).
[CrossRef]

Opt. Express (4)

Phys. Rev. (1)

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

Phys. Rev. B (2)

M. M. J. Treacy, "Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings," Phys. Rev. B 66,195105-11 (2002).
[CrossRef]

Ph. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, "Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures," Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

Ph. Lalanne, J. P. Hugonin, and J. C. Rodier, "Theory of Surface Plasmon Generation at Nanoslit Apertures," Phys. Rev. Lett. 95, 263902 (2005).
[CrossRef]

Proc. Phys. Soc. London (1)

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 (1)

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

Other (9)

E. Noponen, "Electromagnetic Theory of Diffractive Optics," dissertation, Dept. of Technical Physics, Helsinki University of Technology, Finland (1994).

P. Edward, Handbook of optical constants of solids, 1st edition, Academic press, 1997.

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

J. D. Jackson, Classical Electrodynamics, Chapter 8, 3rd edition, Wiley, New York, 1999.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall, London, 1983.

R. N. Bracewell, The Fourier Transform and its Applications, McGraw-Hill, New York, 1978.

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

J. Bravo-Abad, L. Martín-Moreno, F. J. García-Vidal, "Transmission properties of a single metallic slit: from the subwavelength regime to the geometrical-optics limit," Phys. Rev. E 69, 26601(6) (2004).
[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]

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Figures (9)

Fig. 1.
Fig. 1.

A plane-wave illuminates, from the top, a semi-infinite array of slits in a metallic host (silver). The plane of incidence is yz, the vacuum wavelength of the incident beam is λ o=1.0µm, and the permittivity of silver at this frequency is εm =-48.8+i2.99. The slit width is fixed at w=0.1µm, but the array periodicity p is an adjustable parameter of our calculations (0.4µm<p<3.2µm). Only TM modes (Hx , Ey , Ez ) are considered.

Fig. 2.
Fig. 2.

Locations of σz , σys and σym in the complex-plane for mode-numbers 2 through 30, corresponding to a slit array having p=0.9µm, w=0.1µm at λ o=1.0µm, normal incidence. The arrows point in the direction of increasing mode-number, from 2 to 30. The first mode, which is the only guided mode in this system, is off the scale and, therefore, not shown (σz1=1.21+0.0066i, σys1=-0.012+0.68i, σym1=0.21+7.1i). The few abnormalities (i.e., departures from a smooth path) occur at regular intervals, with their Imag[σz ] separated by ~λ o/w. Abnormal modes are produced by lateral Fabry-Perot-like resonances within the slits.

Fig. 3.
Fig. 3.

The first ten modes of a slit array in a semi-infinite silver host (p=0.9µm, w=0.1µm, λ o=1.0µm, normal incidence). The magnitude of Hx is shown on the left, that of Ey on the right. The first mode (shown at the top) is the guided mode through the empty slits (εs =1.0); all the other modes are highly damped along the z-axis. The mode profiles are similar to those computed by Treacy based on the RCWA method [2].

Fig. 4.
Fig. 4.

Inner products of the first 60 modes for Hx (left) and Ey (right). While the Hx -fields of these modes are nearly pairwise orthogonal, the Ey -fields have substantial overlap with their neighboring modes. Each mode is doubly normalized, i.e., ∫ |Hxn (y, 0)|2dy=∫ |Eyn (y, 0)|2dy=1.

Fig. 5.
Fig. 5.

Profiles of Hx and Ey across a full period of the slit array (p=0.9µm, w=0.1µm, λ o=1.0µm, normal incidence), with N=80 modes used to reduce the mismatch at the interface. Excellent agreement is observed between the fields just above the surface (black for Hx , red for Ey ) and those just beneath the surface (green for Hx , blue for Ey ).

Fig. 6.
Fig. 6.

(a) Magnitudes of the first five Bloch modes of the slit array (CT1 ,…, CT5 ) as functions of the total number of modes N used to minimize the mismatch between Ey and Hx across the z=0 interface. (b) Magnitudes of the first 25 Bloch modes of the slit array, with N=80 modes used to minimize the mismatch. (p=0.9 µm, w=0.1µm, λ o=1.0 µm, normal incidence.)

Fig. 7.
Fig. 7.

Transmitted optical power (normalized by the incident power) versus the period p of a slit array in silver over the range p=0.4-3.2 µm (w=0.1 µm, λ o=1.0 µm, normal incidence). The blue curve represents the transmissivity T 1 of the guided mode only (evaluated at z=0+), whereas the red curve shows the total transmissivity T when all modes up to and including N=80 are considered. The difference between the two curves is the optical power absorbed near the entrance facet. Three sets of anomalies appear at and around p=λ o, 2λ o, and 3λ o.

Fig. 8.
Fig. 8.

Magnitudes of the tangential fields beneath the surface (Hx on the left, Ey on the right) across a full period p for four different slit arrays: from top to bottom, p=λspp , λ o, 2λspp , 2λ o; in all cases w=0.1 µm, and λ o=1.0 µm at normal incidence. The mode amplitudes are computed using a total of N=80 modes in each case. The depicted profiles show the superposition of all 80 modes (blue), as well as partial reconstructions containing only the first 3 modes (green), or the first 4 modes (red).

Fig. 9.
Fig. 9.

Plots of total transmission efficiency T, as well as reflection efficiencies Ro , R ±1 of the 0th and ±1st-order diffracted beams, as functions of the wavelength λ o. The array periodicity and slit-width are fixed at p=1.2µm, w=0.1µm, respectively. The incidence angle is θ=0° (red), θ=2° (blue), and θ=4° (black).

Tables (2)

Tables Icon

Table I. Magnitudes of the first five modes (both above and below the z=0 interface) in the vicinity of p=λ o, corresponding to the first anomaly depicted in Fig. 7 (w=0.1 µm, λ o=1.0 µm, normal incidence).

Tables Icon

Table II. Magnitudes of the first five modes (both above and below the z=0 interface) in the vicinity of p=2λ o, corresponding to the second anomaly depicted in Fig. 7 (w=0.1 µm, λ o=1.0 µm, normal incidence).

Equations (21)

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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
E yT n = { 1 i ω ε s ε 0 H xT n ( y , z ) z , w 2 < y < w 2 1 i ω ε m ε 0 H xT n ( y , z ) z , w 2 p y w 2
E zT n = { 1 i ω ε s ε 0 H xT n ( y , z ) y , w 2 < y < w 2 1 i ω ε m ε 0 H xT n ( y , z ) y , w 2 p y w 2
{ H xT n ( y + p , z ) = Λ H xT n ( y , z ) E zT n ( y + p , z ) = Λ E zT n ( y , z )
{ H xT n y = w 2 + = H xT n y = w 2 E zT n y = w 2 + = E zT n y = w 2 H xT n y = w 2 = Λ H xT n y = ( w 2 p ) + E zT n y = w 2 = Λ E zT n y = ( w 2 p ) +
( 1 a b 1 σ ys ε s a σ ys ε s b σ ym ε m σ ym ε m a 1 Λ b Λ a σ ys ε s σ ys ε s Λ σ ym ε m b Λ σ ym ε m ) ( h 1 s h 2 s h 1 m h 2 m ) = 0
( a 2 1 ) ( b 2 1 ) ( ε m 2 σ ys 2 + ε s 2 σ ym 2 ) + 2 ε s ε m [ ( a 2 + 1 ) ( b 2 + 1 ) 2 ab ( Λ + Λ 1 ) ] σ ys σ ym = 0
ε m σ ys ε s σ ym = ( a + 1 ) ( 1 b ) ( a 1 ) ( 1 + b )
H xR n ( y , z ) = exp [ i k 0 ( σ z n z + σ y n y ) ]
H xR n ( y , z ) = exp [ i k 0 ( σ z n z + σ y n y ) ] + exp [ i k 0 ( σ z n z σ y n y ) ]
H xI + n = 1 N C R n H xR n = n = 1 N C T n H xT n
E yI + n = 1 N C R n H yR n = n = 1 N C T n H yT n
H xI n = 1 2 N C n H x n = 0
E yI n = 1 2 N C n H y n = 0
Error = p 2 p 2 { Z 0 2 H xI n = 1 2 N C n H x 2 2 + E yI n = 1 2 N C n E y n 2 } d y
n = 1 2 N C n p 2 p 2 ( Z 0 2 H x j * H x n + E y j * E y n ) dy = p 2 p 2 ( Z 0 2 H x j * H xI + E y j * E yI ) dy
R n = p 2 p 2 Re [ C R n E yR n × ( C R n H xR n ) * ] dy p 2 p 2 Re ( E yI × H xI * ) dy
T n = p 2 p 2 Re [ C T n E yT n × ( C T n H xT n ) * ] dy p 2 p 2 Re ( E yI × H xI * ) dy
T = p 2 p 2 Re [ n = 1 N C T n E yT n × n = 1 N ( C T n H xT n ) * ] dy p 2 p 2 Re ( E yI × H xI * ) dy
sin θ + m ( λ Wood p ) = ± 1
sin θ + m ( λ spp p ) = ± n spp ( λ )

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