Abstract

The surface plasmon self-interference excited by a strongly focused, linearly polarized vortex beam at off-axis illumination in a paraxial regime is analytically studied. The off-axis excitation is investigated using a geometrical model. The combination of an angular spectrum representation and homogeneous transformation is applied to derive the integral expressions of the surface plasmon polariton fields for off-axis directions both parallel and perpendicular to polarization plane, and an off-axis convergence angle is used to compute the integral. The surface plasmon excitation is represented by the relative peak intensity of the longitudinal field, while its standing wave is characterized by the full width at half-maximum of the transmitted field intensity distribution profile. Both models consistently show that even in ideal Gaussian microscopic imaging systems, self-interference degradation exists. When the off-axis angle increases, the surface plasmon interference disappears and the fields detune out of surface plasmon resonance.

© 2017 Optical Society of America

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References

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  1. H. Raether, Surface-plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).
  2. 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(6668), 667–669 (1998).
    [Crossref]
  3. M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6(3), 130–132 (2012).
    [Crossref]
  4. Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  8. B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
    [Crossref]
  9. P. S. Tan, X. C. Yuan, J. Lin, Q. Wang, and R. E. Burge, “Analysis of surface plasmon interference pattern formed by optical vortex beams,” Opt. Express 16(22), 18451–18456 (2008).
    [Crossref] [PubMed]
  10. A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J. C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32(17), 2535–2537 (2007).
    [Crossref] [PubMed]
  11. W. Zou, P. Huang, W. Ma, and F. Guo, “Theoretical analysis of obliquely excited surface plasmon self-interference,” Opt. Express 21(15), 18572–18581 (2013).
    [Crossref] [PubMed]
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    [Crossref]
  13. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
    [Crossref]
  14. L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge University, 2006).
  15. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2005).

2013 (2)

2012 (1)

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6(3), 130–132 (2012).
[Crossref]

2010 (1)

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

2005 (1)

Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
[Crossref] [PubMed]

2000 (1)

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(6668), 667–669 (1998).
[Crossref]

1959 (2)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

Bouhelier, A.

Bruyant, A.

Burge, R. E.

Carminati, R.

R. Carminati, “Surface plasmons: A probe for graphene electronics,” Nat. Nanotechnol. 8(11), 802–803 (2013).
[Crossref] [PubMed]

Chung, E.

Colas des Francs, G.

Cragg, G. E.

Dereux, A.

Ebbesen, T. W.

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(6668), 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(6668), 667–669 (1998).
[Crossref]

Green, M. A.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6(3), 130–132 (2012).
[Crossref]

Guo, F.

Huang, C.

Huang, P.

Ignatovich, F.

Kim, D.

Kim, H.

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

Kim, S.

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

Lee, B.

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

Lezec, H. J.

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(6668), 667–669 (1998).
[Crossref]

Lim, Y.

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

Lin, J.

Liu, Z. W.

Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
[Crossref] [PubMed]

Ma, W.

Novotny, L.

Pillai, S.

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6(3), 130–132 (2012).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

So, P. T. C.

Tan, P. S.

Thio, T.

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(6668), 667–669 (1998).
[Crossref]

Wang, Q.

Weeber, J. C.

Wei, Q. H.

Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
[Crossref] [PubMed]

Wiederrecht, G. P.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

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(6668), 667–669 (1998).
[Crossref]

Yuan, X. C.

Zhang, X.

Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
[Crossref] [PubMed]

Zou, W.

Nano Lett. (1)

Z. W. Liu, Q. H. Wei, and X. Zhang, “Surface plasmon interference nanolithography,” Nano Lett. 5(5), 957–961 (2005).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

R. Carminati, “Surface plasmons: A probe for graphene electronics,” Nat. Nanotechnol. 8(11), 802–803 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

M. A. Green and S. Pillai, “Harnessing plasmonics for solar cells,” Nat. Photonics 6(3), 130–132 (2012).
[Crossref]

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(6668), 667–669 (1998).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

Prog. Quantum Electron. (1)

B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quantum Electron. 34(2), 47–87 (2010).
[Crossref]

Other (3)

H. Raether, Surface-plasmons on Smooth and Rough Surfaces and on Grating, Springer Tracts in Modern Physics (Springer Berlin, 1988).

L. Novotny and B. Hetch, Principle of Nano-optics (Cambridge University, 2006).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2005).

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Figures (8)

Fig. 1
Fig. 1 Schematic diagrams of two abnormal SPP excitation conditions (a) oblique excitation, (b) off-axis excitation.

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Fig. 2
Fig. 2 Geometrical representation of SPP excitation by an off-axis and tightly focused beam

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Fig. 3
Fig. 3 (a) Three-dimensional plot of an off-axially focused light beam converging towards geometric focus O′ on the glass/metal interface located at z = 0. The solid line cone represents the off-axially focused beam, while the dash line cone represents the light rays with incident angle θsp. (b) Top-view diagram illustrating the generation of off-axially excited SPP self-interference. The line segment AB is the diameter of circle F′. A1 and B1 are the points of intersection of circles F and F′, A2 and B2 are symmetric to A1 and B1, respectively, with respect to AB. Obviously, A2 and B2 are points on circle F′.

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Fig. 4
Fig. 4 Plots of θ ′ ∥ ( γ , ϕ ′ ) as a function of γ and ϕ ′ . For this calculation, we used n1 = 1.519 and NA = 1.4.

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Fig. 5
Fig. 5 SPP self-interference pattern maps on the Au film surface when viewed in -z direction. The vortex beam is focused at an off-axis angle of 10° in the direction (a) parallel, and (b) perpendicular to the polarization plane.

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Fig. 6
Fig. 6 FWHM vs. off-axis angle. In the simulation, the SPPs are excited on gold film using a 532 nm vortex beam focused by a lens of different NA values in the deviating direction (a) parallel and (b) perpendicular to the polarization plane. (c) SPPs intensity profile along x-axis at on-axis focusing and normal incidence (NA = 1.4).

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Fig. 7
Fig. 7 Relative peak intensity IZ/I0 vs. off-axis angle. In the simulation, the SPPs are excited on an Au film using a 532 nm vortex beam focused by a lens of different NA values in the deviating direction (a) parallel and (b) perpendicular to the polarization plane.

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Fig. 8
Fig. 8 Relative peak intensity curves in the neighborhood of γoad (NA = 1.4).

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Tables (1)

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Table 1 Off-axis detuning angle for SPP excitation on Au film by a lens of different NA values

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Equations (24)

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tan γ c o a = F F ′ / f   ,         tan θ max = B F / f   ,       tan θ s p = F ′ B / f .
γ c o a = arc tan [ ( tan 2 θ max − tan 2 θ s p )   1 / 2 ] .
{ x 2 + y 2 = a 2 z = − f .
[ x − t y z ] = R T [ x ′ y ′ z ′ ] , [ k x k y k z ] = R T [ k x ′ k y ′ k z ′ ] .
R = [ cos γ 0 − sin γ 0 1 0 sin γ 0 cos γ ] .
{ x ′ = f tan θ max cos γ cos ϕ ′ y ′ = f tan θ max sin ϕ ′ z ′ = f tan θ max sin γ cos ϕ ′ − f sec γ .
O ′ P = x ′ n x ′ + y ′ n y ′ + ( x ′ tan γ − f sec γ ) n z ′ .
O ′ F =- f sec γ n z ′ .
θ ′ ∥ ( γ , ϕ ′ ) = arc cos ( O ′ F ⋅ O ′ P | O ′ F | | O ′ P | ) .
θ ′ ∥ ( γ , ϕ ′ ) = arc cos ( 1 − tan θ max sin γ cos γ cos ϕ ′ 1 − 2 tan θ max sin γ cos γ cos ϕ ′ + tan 2 θ max cos 2 γ ) .
R = [ 1 0 0 0 cos γ sin γ 0 − sin γ cos γ ] .
θ ′ ⊥ ( γ , ϕ ′ ) = θ ′ ∥ ( γ , ϕ ′ + π 2 ) .
E 3 ( x , y , z ) = i f e − i k 1 f 2 π ∬ k x k y t p ( k z j ) ( E i n c ⋅ n ρ ) n θ k z 1 / k 1 k z 1 exp [ i ( k x x + k y y + k z 3 z ) ] d k x d k y .
t p ( θ ′ , ϕ ′ , γ ) = 4 exp [ i   ( k z 2 − k z 3 ) d ] ( 1 + p 12 ) ( 1 + p 23 ) [ ( 1 + r 12 r 23 exp ( i   2 k z 2 d ) ] , k z j = k j 2 − ( k x 2 + k y 2 ) ,       p i j = ε i k z j ε j k z i ,       r i j = 1 − p i j 1 + p i j .
E i n c ( k x ′ , k y ′ ) = E i n c ( k x ′ , k y ′ ) x ′ = E 0 exp [ −   k x ′ 2 + k y ′ 2 k 1 2 sin 2 θ max   +   i l ϕ ′ ] x ′ .
E i n c ⋅ n ρ = E i n c ( k x ′ , k y ′ ) k x cos γ k x 2 + k y 2 .
| ∂ ( k x , k y ) ∂ ( k x ′ , k y ′ ) | = cos γ − k x ′ k z ′ 1 sin γ .
E z 3 ( x , y , d ) = i f e − i k 1 f 2 π ∬ k x ′ k y ′ t p [ k z j ( k x ′ , k y ′ ) ] E i n c ( k x ′ , k y ′ ) exp [ i k z 3 ( k x ′ , k y ′ ) d ] ( k x ′ cos γ + k z ′ 1 sin γ ) × exp { i [ ( k x ′ cos γ + k z ′ 1 sin γ ) ( x − t ) + k y ′ y ] } k z ′ 1 / k 1 k 1 k z ′ 1 cos γ d k x ′ d k y ′ .
E z 3 ( x , y , d ) ∝ ∫ 0 2 π   ∫ 0 θ ′ ∥ t p [ k z j ( θ ′ , ϕ ′ ) ] E i n c ( θ ′ , ϕ ′ , γ ) exp [ i k z 3 ( θ ′ , ϕ ′ ) d ] × ( sin θ ′ cos ϕ ′ cos γ + cos θ ′ sin γ )   exp { i k 1 [ ( sin θ ′ cos ϕ ′ cos γ + cos θ ′ sin γ ) × ( x − f tan γ ) + sin θ ′ sin ϕ ′ y ] }   ( cos θ ′ ) 1 / 2 sin θ ′ cos γ d θ ′ d ϕ ′ .
E i n c ⋅ n ρ = E i n c ( k x ′ , k y ′ ) k x k x 2 + k y 2 .
| ∂ ( k x , k y ) ∂ ( k x ′ , k y ′ ) | = cos γ + k y ′ k z ′ 1 sin γ .
E z 3 ( x , y , d ) = i f e − i k 1 f 2 π ∬ k x k y t p [ k z j ( k x ′ , k y ′ ) ] E i n c ( k x ′ , k y ′ ) exp [ i k z 3 ( k x ′ , k y ′ ) d ] × k x ′ k z ′ 1 / k 1 k 1 k z ′ 1 exp { i [ k x ′ x + ( k y ′ cos γ − k z ′ 1 sin γ ) ( y − t ) ] } d k x ′ d k y ′ .
E z 3 ( x , y , d ) ∝   ∫ 0 2 π   ∫ 0 θ ′ ⊥ t p [ k z j ( θ ′ , ϕ ′ ) ] E i n c ( θ ′ , ϕ ′ , γ ) exp [ i k z 3 ( θ ′ , ϕ ′ ) d ] × exp { i k 1 [ sin θ ′ cos ϕ ′ x + ( sin θ ′ sin ϕ ′ cos γ − cos θ ′ sin γ ) ( y + f tan γ ) ] }   × ( cos θ ′ ) 1 / 2 sin 2 θ ′ cos ϕ ′ d θ ′ d ϕ ′ .
θ ′ ( γ o a d , 0 ) = γ o a d − θ s p .

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