Abstract

We demonstrate by a rigorous theoretical calculation the transient Surface Plasmon Polariton (SPP) mode, which is launched by a beam impinging onto a sub-wavelength real metallic slit in addition to the conventional long-range SPP. Different from the previous works, we find a direct closed-form solution of the Maxwell’s equations by the Sommerfeld branch-cut integrals without approximation. The transient wave is a SPP with a complex-valued envelope, expressed as the exponential integral. Its rapid damping may be asymptotically approximated as -ln(x), 1/x 1/2 and 1/x, respectively, depending on the distance range away from the slit. The transit SPP may be considered as a cylindrical wave, which is radiated from the slit, takes the SPP propagation constant at the interface and propagates with additional drop due to the loss of energy pumped to the SPP.

© 2008 Optical Society of America

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  1. G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
    [CrossRef]
  2. H. J. Lezec and T. Thio, "Diffracted evansecent wave model for enhanced and suppressed optical transmission through subwavelength hole arrays," Opt. Express 12,3629-3651 (2004).
    [CrossRef] [PubMed]
  3. H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
    [CrossRef] [PubMed]
  4. B. Ung and Y. Sheng, "Interference of surface waves in a metallic subwavelength slit," Opt. Express 15, 1182-1190 (2007).
    [CrossRef] [PubMed]
  5. M. W. Kowarz, "Homogeneous and evanescent contributions in scalar near-field diffraction," Appl. Opt. 34, 3055-3063 (1995).
    [CrossRef] [PubMed]
  6. P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nature Phys. 2, 551-556 (2006).
    [CrossRef]
  7. B. Ung and Y. Sheng, "Optical surface waves over metallo-dielectric nanostructures: Sommerfeld integrals revisited," Opt. Express 16, 9073-9086 (2008).
    [CrossRef] [PubMed]
  8. G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
    [CrossRef]
  9. L. Chen, J. T. Robinson, and M. Lipson, "Role of radiation and surface plasmon polaritons in the optical interaction between a nano-slit and a nano-groove on a metal surface," Opt. Express 14, 12629-12636 (2006)
    [CrossRef] [PubMed]
  10. See Section 4 in Ref [8].
  11. L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
    [CrossRef] [PubMed]
  12. 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. R. E. Collin, "Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies," IEEE Ant. Prop. Mag. 46, 64-79 (2004).
    [CrossRef]
  14. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic waves: theory and applications to communications, geophysical exploration and remote sensing (Springer-Verlag, New York, 1992)
    [PubMed]

2008 (1)

2007 (3)

G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
[CrossRef]

B. Ung and Y. Sheng, "Interference of surface waves in a metallic subwavelength slit," Opt. Express 15, 1182-1190 (2007).
[CrossRef] [PubMed]

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

2006 (3)

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

L. Chen, J. T. Robinson, and M. Lipson, "Role of radiation and surface plasmon polaritons in the optical interaction between a nano-slit and a nano-groove on a metal surface," Opt. Express 14, 12629-12636 (2006)
[CrossRef] [PubMed]

P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nature Phys. 2, 551-556 (2006).
[CrossRef]

2005 (1)

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

2004 (3)

1995 (1)

Aigouy, L.

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

Alkemade, P.F.A.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Alloschery, O.

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Blok, H.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Chen, L.

Collin, R. E.

R. E. Collin, "Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies," IEEE Ant. Prop. Mag. 46, 64-79 (2004).
[CrossRef]

Dubois, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Eliel, E. R.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Gay, G.

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Gbur, G.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Hooft, G.W.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Hugonin, J. P.

P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nature Phys. 2, 551-556 (2006).
[CrossRef]

Hugonin, J.P.

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

Julié, G.

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

Kowarz, M. W.

Kuzmin, N.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Lalanne, P.

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nature Phys. 2, 551-556 (2006).
[CrossRef]

Lenstra, D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Lévêque, G.

G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
[CrossRef]

Lezec, H. J.

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

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

Lipson, M.

Martin, O. J. F.

G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
[CrossRef]

Mortier, M.

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

O'Dwyer, C.

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Robinson, J. T.

Schouten, H. F.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Sheng, Y.

Thio, T.

Ung, B.

Viaris De Lesegno, B.

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Visser, T. D.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

Weiner, J.

G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
[CrossRef]

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Xie,

Appl. Opt. (1)

IEEE Ant. Prop. Mag. (1)

R. E. Collin, "Hertzian dipole radiating over a lossy earth or sea: some early and late 20th-century controversies," IEEE Ant. Prop. Mag. 46, 64-79 (2004).
[CrossRef]

Nature Phys. (2)

P. Lalanne and J. P. Hugonin, "Interaction between optical nano-objects at metallo-dielectric interfaces," Nature Phys. 2, 551-556 (2006).
[CrossRef]

G. Gay, O. Alloschery, B. Viaris De Lesegno, C. O'Dwyer, J. Weiner, and H. J. Lezec, "The optical response of nanostructured surfaces and the composite diffracted evanescent wave model," Nature Phys. 2, 262-267 (2006).
[CrossRef]

Opt. Express (5)

Phys. Rev. B (1)

G. Lévêque, O. J. F. Martin, and J. Weiner, "Transient behavior of surface plasmon polaritons scattered at a subwavelength groove," Phys. Rev. B 76, 155418 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P.F.A. Alkemade, H. Blok, G.W. Hooft, D. Lenstra, and E. R. Eliel, "Plasmon-assisted two-slit transmission: Young’s experiment revisited," Phys. Rev. Lett. 94, 053901 (2005).
[CrossRef] [PubMed]

L. Aigouy, P. Lalanne, J.P. Hugonin, G. Julié, and M. Mortier, "Near-Field Analysis of Surface Waves Launched at Nanoslit Apertures," Phys. Rev. Lett. 98, 153902 (2007).
[CrossRef] [PubMed]

Other (2)

R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic waves: theory and applications to communications, geophysical exploration and remote sensing (Springer-Verlag, New York, 1992)
[PubMed]

See Section 4 in Ref [8].

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Figures (6)

Fig. 1.
Fig. 1.

Horizontal electrical dipole induced by incident TM field at a real-metal slit and the equivalent magnetic source line J*m

Fig. 2.
Fig. 2.

Contour path C in complex plane β for integral in Eq. (2).

Fig. 3.
Fig. 3.

Contour C’ in the complex plane t for integral in Eq. (8).

Fig. 4.
Fig. 4.

Path for computing the last integral in Eq. (9).

Fig. 5.
Fig. 5.

(color online) a) Transient SPP envelope E1(ikspx) with Euler constant γ~0.5772 and b) Phase of E1(ikspx); for sliver slit in air with εm =-41.23+2.82i at λ=1 µm.

Fig. 6.
Fig. 6.

(color online) Amplitude of the transient SPP (black) for a) λ=1 µm and b) λ=9 µm. Curves damp as -γ-ln(x) (green), 1/x1/2 (blue) and 1/x (red) in the three ranges of x, for a sliver slit in air.

Equations (37)

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( y 2 + γ j 2 ) H ˜ j = i ω ε j J m * δ ( y + d ) ,
H z ( x ) = ω J m * 2 π ε m ε d exp ( i k 0 β x ) ε d ε m β 2 + ε m ε d β 2 d β ,
β p = ε m ε d ( ε m + ε d ) = k sp k 0 ,
H sp = i 2 π e i k sp x ( ε d ε m ) 3 2 ε d 2 + ε m 2 ,
H z ( x ) = ω J m * 2 π ( H sp + Γ m f ( β ) d β + Γ d f ( β ) d β ) ,
f ( t ) = i ε m ε d exp ( i k 0 x ε m + it ) 2 ε m + it ( ε m ε d ( ε m + it ) + ε d it ) .
Γ m f ( β ) d β = 0 f + ( t ) dt + 0 f ( t exp ( 2 π i ) ) dt ,
Γ m f ( β ) d β = ε d e i π 4 k sp 2 ε m ε d k sp 2 k 0 2 0 exp ( i k 0 x ε m + it ) t ε m + it ( it + ε m 2 ( ε m + ε d ) ) dt .
0 φ ( t ) dt = i π e i π 4 ε m ε d e i k sp x 1 2 Γ m ( t ) φ ( t ) dt .
φ ( τ ) = 2 i e i π 4 τ 2 ε m τ 2 k sp 2 k 0 2 e i k 0 x τ .
2 0 φ ( τ ) d τ = 4 i e i π 4 k 0 0 e sk f 1 ( k ) f 2 ( k ) dk ,
{ f 1 ( s ) f 2 ( s ) } = 1 2 π i α i α + i F 1 ( s ) F 2 ( p s ) ds ,
F 1 ( s ) { f 1 ( k ) } = i π k m 2 s ( H 1 ( i k s s ) Y 1 ( i k m s ) ) .
{ f 1 ( s ) f 2 ( s ) } = lim s 0 d ds F 2 ( p s ) = 1 2 ( 0 e ikx k k sp dk + 0 e ikx k k sp dk ) ,
0 e ikx k k sp dk = e i k sp x E 1 ( i k sp x ) , x 0 ,
E 1 (z)= z ( e t /t)dt.
H z ( x ) = i ω J m * ε m ε d ( ( ε m ε d ) 1 2 ε d 2 + ε m 2 + E 1 ( i k sp x ) 2 π ( ε m + ε d ) ) e i k sp x ,
e i k sp x E 1 ( i ( k sp k 0 ) x ) .
× E j = J m * μ 0 t H j ;
× H j = ε j t E j ,
x ( E y ) j y ( E x ) j = J m * δ ( x ) δ ( y + d ) μ 0 t ( H z ) j ;
y ( H z ) j = ε j t ( E x ) j ;
x ( H z ) j = ε j t ( E y ) j .
A j ( x , y ) = 1 2 π A ̅ j ( k 0 β , y ) e i k 0 β x d ( k 0 β ) ,
i k 0 β ( E ˜ y ) j y ( E ˜ x ) j = J m * δ ( y + d ) + i ω μ 0 ( H ˜ z ) j ;
y ( H ˜ z ) j = i ω ε j ( E ˜ x ) j ;
i k 0 β ( H ˜ z ) j = i ω ε j ( E ˜ y ) j .
H ˜ 1 ( h ) = C 1 e i γ 1 y + C 2 e i γ 1 y .
H ˜ 1 ( p ) = ω ε 1 J m * 2 γ 1 e i γ 1 y + d .
H ˜ 1 = C 1 e i γ 1 y + C 2 e i γ 1 y ω ε 1 J m * 2 γ 1 e i γ 1 y + d .
H ˜ 2 = C 3 e i γ 2 y + C 4 e i γ 2 y .
H ˜ 1 = C 2 e i γ 1 y ω ε 1 J m * 2 γ 1 e i γ 1 y + d ;
H ˜ 2 = C 2 e i γ 2 y .
H ˜ 1 y = 0 = H ˜ 2 y = 0 ;
( E ˜ x ) 1 y = 0 = ( E ˜ x ) 2 y = 0 .
C 3 = C 2 ω ε 1 J m * 2 γ 1 e i γ 1 d .
C 2 = ω J m * 2 + ω J m * ε 1 γ 2 2 ε 2 γ 1 γ 1 ε 1 + γ 2 ε 2 e i γ 1 d .

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