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

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  1. 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]
  2. R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998).
    [CrossRef]
  3. 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]
  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," Nat. Phys. 264, 262 - 67 (2006).
    [CrossRef]
  5. 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]
  6. 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]
  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, Pure Appl. Opt.  2, 48-51 (2000).
    [CrossRef]
  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 72, 075405 (2005).
    [CrossRef]
  9. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1986).
  10. 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]
  11. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., (Artech House, 2000).
  12. 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]
  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]

2006 (2)

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]

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]

2005 (1)

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]

2004 (1)

2002 (1)

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]

2000 (1)

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]

1998 (3)

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]

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]

1986 (1)

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]

Alloschery, O.

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]

Astilean, S.

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]

Atwater, H. A.

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]

Averitt, R. D.

R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998).
[CrossRef]

Burke, J. J.

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]

de Lesegno, B. V.

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]

Dionne, J. A.

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]

Ebbesen, T. W.

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]

Gay, G.

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]

Ghaemi, H. F.

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]

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]

Grupp, D. E.

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]

Halas, N. J.

R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998).
[CrossRef]

Hugonin, J. P.

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]

Lalanne, Ph.

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]

Lezec, H. J.

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]

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]

Mansuripur, M.

Mock, J. J.

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]

Moller, K. D.

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]

Moloney, J. V.

O'Dwyer, C.

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]

Oldenburg, S. J.

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]

Palamaru, M.

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]

Polman, A.

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]

Schultz, D. A.

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]

Schultz, S.

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]

Smith, D. R.

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]

Stegeman, G. I.

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]

Sweatlock, L. A.

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]

Tamir, T.

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]

Thio, T.

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]

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]

Weiner, J.

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]

Westcott, S. L.

R. D. Averitt, S. L. Westcott, and N. J. Halas, "Ultrafast electron dynamics in gold nanoshells," Phys. Rev. B 58, R10203-R10206 (1998).
[CrossRef]

Wolff, P. A.

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]

Xie, Y.

Zakharian, A. R.

J. Opt. A, (1)

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]

Nano Lett. (1)

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]

Nat. Phys. (1)

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]

Nature (1)

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]

Opt. Express (2)

Phys. Rev. B (4)

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]

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. 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. F. Ghaemi, T. Thio, and D. E. Grupp, "Surface plasmons enhance optical transmission through subwavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[CrossRef]

Other (3)

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]

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, Berlin, 1986).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed., (Artech House, 2000).

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Figures (13)

Fig. 1.
Fig. 1.

Slab of thickness w and dielectric constant ε 2, sandwiched between two homogeneous, semi-infinite media of dielectric constant ε 1. An electromagnetic mode of the structure consists of two (generally inhomogeneous) plane-waves within the slab and a single (inhomogeneous) plane-wave in each of the surrounding media. Continuity of the fields at z = ± 1 2 w w requires that ky = k o σy be the same for all these plane-waves. Although the polarization state of the mode can, in general, be either TE or TM, only TM modes are considered in this paper. The magnetic field, therefore, has a single component Hx along the x-axis, while the electric field has two components (Ey , Ez ) in the yz-plane. Throughout the paper λ o = 650 nm and the metallic medium is silver, having ε = -19.6224 + 0.443i (corresponding to n + i k = 0.05 + 4.43i).

Fig. 2.
Fig. 2.

(a) The SPP’s E-fields originate on positive charges and terminate on negative ones. Continuity of H and a negative ε metal ensure the continuity of D⊥, while E becomes continuous when σy = σspp . (b) A physical impossibility, since the divergence-free nature of the H-field requires H to have opposite directions above and below the surface, thus prohibiting the continuity of H at the boundary.

Fig. 3.
Fig. 3.

(a) Real and imaginary parts of ε(ω) for a metal that obeys the Drude model in the frequency range from near IR to UV; the inset is a close-up of the right tail of the curves. Re[ε(ωp )] = 0 at the plasma frequency ωp , which corresponds to λ o = 143 nm, whereas [ ε ( ω p 2 ) ] = 1 , corresponding to λ o = 202 nm. (b) Plots of the real and imaginary parts of σ y ( ω ) = ε ( ω ) [ 1 + ε ( ω ) ] .

Fig. 4.
Fig. 4.

Group velocity Vg (ω), normalized by the speed of light c, in the frequency range (a) below ω p 2 and (b) above ωp . Also shown are the coefficients [ωσ y (i)(ω)]′, [ωσ y (r)(ω)]″, [ωσ y (i)(ω)]″, which appear in Eq. (9) and cause pulse distortion during propagation.

Fig. 5.
Fig. 5.

A Gaussian beam of wavelength λ o is focused at the entrance facet of a semi-infinite slit (width = w, dielectric constant = ε 2) in a metallic host having dielectric constant ε 1. (In this 2-dimensional system the beam is focused through a cylindrical lens; its shape, therefore, does not vary along the x-axis.) The beam is linearly polarized, having H-field along x and E-field components (Ey , Ez ) confined to the plane of incidence. In our FDTD simulations, the incident beam (λ o = 650 nm, amplitude FWHM = 4 μm) is sourced at y = -10 nm, just before the entrance facet of the slit.

Fig. 6.
Fig. 6.

Left to right: Instantaneous field profiles Hx , Ev , Ez show a guided mode propagating down the slit (w = 200 nm, ε 2 = 1.0) and a reflected beam propagating backward in the incidence space. Long-range SPPs excited on the front facet of the metallic host (both above and below the slit) are clearly visible in the Hx and Ey plots. (The incident beam, sourced at Δy = 10 nm before the entrance facet, is largely absent from these plots, hence the absence of interference fringes between the incident and reflected beams.)

Fig. 7.
Fig. 7.

Left to right: Poynting vector components Sy , Sz , and magnitude |S| in and around the 200 nm-wide slit simulated in Fig. 6. The incident beam having been removed, only reflected and transmitted beams appear in these pictures.

Fig. 8.
Fig. 8.

Profiles of Hx (z) at Δy = 10 nm beneath the metal surface. |Hx | (green) indicates the field strength, while Re[Hx ] (black) reveals phase variations across the surface. (a) Beyond |z| ∼ 5 μm the field amplitude |Hx | becomes nearly constant, indicating the presence of a low-loss SPP propagating away from the slit. Oscillations of |Hx | in the immediate vicinity of the slit are caused by interference between the small fraction of the Gaussian beam that penetrates the metallic surface and the SPP launched at the slit. (b) Subtracting from the total Hx at Δy = 10 nm, an estimated profile of the Gaussian beam (obtained by setting w = 0 in a separate simulation), reveals the dominance of SPP at |z| > 1 μm

Fig. 9.
Fig. 9.

The SPPs of Fig. 8 detach from the slit and travel away when the incident beam is a short, Gaussian pulse (FWHM width τ= 20 fs). Shown are profiles of Re[Hx (z)] on one side of the slit at Δy = 20 nm before the metal surface at t = 60, 80, 100 fs after the start of the pulse.

Fig. 10.
Fig. 10.

Strength of the SPP launched on the front facet of the metallic host as function of the slit width (λ o = 650 nm, semi-infinite silver host). The magnitude of Ey ( z = 1 2 w + 6.2 μ m , y = 20 nm ) is taken as the SPP’s strength. The incident Gaussian beam is described in the caption to Fig. 5.

Fig. 11.
Fig. 11.

Same as Fig. 1, except for the existence of counter-propagating waves in the semi-infinite cladding regions above and below the slab waveguide. An electromagnetic mode of the structure thus consists of six (generally inhomogeneous) plane-waves – two within each of the three media. Continuity of the fields at the z = ± 1 2 w boundaries requires that σy be the same for all these plane-waves. We impose the further restriction that σ z1 be real-valued, as any imaginary component of σ z1 causes the beams in the cladding region to grow indefinitely when z → ±∞. Only TM modes having field components (Hx , Ey , Ez ) are considered in this paper.

Fig. 12.
Fig. 12.

Real and imaginary parts of H 1′/H 1 as functions of σ z1. (a) Even modes corresponding to the plus sign in Eq. (12). (b) Odd modes corresponding to the minus sign.

Fig. 13.
Fig. 13.

(a) The function f(z), representing an arbitrary plasmonic wave on the positive z-axis (λ o = 0.65μm, (σspp = 1.1 + 0.03i), rapidly drops to zero when z goes negative. (b) The Fourier transform F(σz ) of f(z) has spatial frequency content confined to the vicinity of σz = Re[σspp ]/λ o. (c, d) The function g(z) and its Fourier transform G(σz ) obtained by flipping F(σz ) to the negative side of the σz -axis, then multiplying it by -1. (e) Combining the positive- and negative-frequency spectra of (b) and (d) yields the function f(z) + g(z), shown here on the positive side of the z-axis only. The main effect of adding negative-frequency terms to the plasmonic function f(z) is seen to be in the neighborhood of the origin; the inset is a close-up of |f(z) + g(z)| around z = 0.

Tables (4)

Tables Icon

Table 1. First few solutions of Eq. (4) for a 50 nm-thick silver slab (λ o = 650 nm, ε1 = 1.0, ε2 = -19.6224 + 0.443i)

Tables Icon

Table 2. Fundamental modes of silver slabs of differing thickness (λ o = 650 nm, ε 1 = 1.0, ε 2 = -19.6224 + 0.443i)

Tables Icon

Table 3. First few solutions of Eq. (4) for a 100 nm-wide slit in a silver host (λo = 650 nm, ε 1 = -19.6224 + 0.443i, ε 2 = 1.0; guided mode in red).

Tables Icon

Table 4. First few TM modes of slits of differing width in a silver host (λ o = 650 nm, ε 1 = -19.6224 + 0.443i, ε 2 = 1.0; guided modes in red)

Equations (18)

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H x y z t = H o exp { i [ k o ( σ y y ± σ z z ) ωt ] }
E y y z t = ( Z o σ z ε ) H x y z t
E z y z t = ( Z o σ y ε ) H x y z t
H x y z = { H 1 exp ( i k o σ y y ) exp [ i k o σ z 1 ( z 1 2 w ) ] ; z + 1 2 w H 2 exp ( i k o σ y y ) [ exp ( i k o σ z 2 z ) ± exp ( i k o σ z 2 z ) ] ; z 1 2 w ± H 1 exp ( i k o σ y y ) exp [ i k o σ z 1 ( z + 1 2 w ) ] ; z 1 2 w
H 2 [ exp ( i k o σ z 2 w 2 ) ± exp ( i k o σ z 2 w 2 ) ] = H 1
Z o H 2 ( σ z 2 ε 2 ) [ exp ( i k o σ z 2 w 2 ) exp ( i k o σ z 2 w 2 ) ] = Z o H 1 ( σ z 1 ε 1 )
ε 1 ε 2 σ y 2 ε 2 ε 1 σ y 2 ε 1 ε 2 σ y 2 + ε 2 ε 1 σ y 2 exp ( i k o ε 2 σ y 2 w ) = ± 1
exp [ k o w σ y ( i ) ] exp [ i k o w σ y ( r ) σ y ( r ) σ y ( r ) ] ± ( ε 1 + ε 2 ) ( ε 1 ε 2 )
σ y ( ω ) = ε 2 ( 1 + ε 2 ) = [ ( ω 2 ω p 2 ) + i γω ] [ ( 2 ω 2 ω p 2 ) + 2 i γω ] .
f y o t = F ( ω ω o ) exp { i ω [ σ y ( ω ) y o c t ] } d ω .
ω σ y ( ω ) ω o σ y ( ω o ) + [ ω σ y ( ω ) ] ( ω ω o ) + 1 2 [ ω σ y ( ω ) ] ′′ ( ω ω o ) 2 .
f y o t exp { k o [ σ y ( i ) ( ω o ) i σ y ( r ) ( ω o ) + i [ ω σ y ( r ) ] ] y o }
× exp { [ [ ω σ y ( i ) ] ( ω ω o ) + 1 2 [ ω σ y ( i ) ] ′′ ( ω ω o ) 2 1 2 i [ ω σ y ( r ) ] ′′ ( ω ω o ) 2 ] y o c } F ( ω ω o )
× exp { [ [ ω σ y ( r ) ] y o c t ] } .
H x y z = { exp ( i k o σ y y ) { H 1 exp [ i k o σ z 1 ( z 1 2 w ) ] + H 1 exp [ i k o σ z 1 ( z 1 2 w ) ] } ; z + 1 2 w H 2 exp ( i k o σ y y ) [ exp ( i k o σ z 2 z ) ± exp ( i k o σ z 2 z ) ] ; z 1 2 w ± exp ( i k o σ y y ) { H 1 exp [ i k o σ z 1 ( z + 1 2 w ) ] + H 1 exp [ i k o σ z 1 ( z + 1 2 w ) ] } ; z 1 2 w
H 2 [ exp ( i k o σ z 2 w 2 ) ± exp ( i k o σ z 2 w 2 ) ] = H 1 + H 1
Z o H 2 ( σ z 2 ε 2 ) [ exp ( i k o σ z 2 w 2 ) exp ( i k o σ z 2 w 2 ) ] = Z o ( σ z 1 ε 1 ) ( H 1 H 1 )
H 1 H 1 = 1 ± [ ( ε 2 σ z 1 ε 1 σ z 1 2 + ε 2 ε 1 ) ( ε 2 σ z 1 + ε 1 σ z 1 2 + ε 2 ε 1 ) ] exp ( i k 0 σ z 1 2 + ε 2 ε 1 w ) [ ( ε 2 σ z 1 ε 1 σ z 1 2 + ε 2 ε 1 ) ( ε 2 σ z 1 + ε 1 σ z 1 2 + ε 2 ε 1 ) ] ± exp ( i k 0 σ z 1 2 + ε 2 ε 1 w )

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