Abstract

We explore electromagnetic coherence lattices in planar polychromatic surface plasmon polariton (SPP) fields. When the SPP constituents are uncorrelated—and thus do not interfere—coherence lattices arise from statistical similarity of the random SPP electromagnetic field. As the SPP correlations become stronger, the coherence lattices fade away, but the lattice structure reemerges in the spectral density of the field. The polarization states of the structured SPP lattice fields are also investigated. Controllable plasmonic coherence and spectral density lattices can find applications in nanophotonics, such as nanoparticle manipulation.

© 2018 Optical Society of America

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References

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2018 (2)

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, Opt. Lett. 43, 1395 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Phys. Rev. A 97, 041801 (2018).
[Crossref]

2017 (1)

2016 (6)

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[Crossref]

A. T. Friberg and T. Setälä, J. Opt. Soc. Am. A 33, 2431 (2016).
[Crossref]

Y. Wang, Y. Xu, X. Feng, P. Zhao, F. Liu, K. Cui, W. Zhang, and Y. Huang, Opt. Lett. 41, 1478 (2016).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, Appl. Phys. Lett. 109, 061107 (2016).
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X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, Opt. Lett. 41, 4182 (2016).
[Crossref]

D. Morrill, D. Li, and D. Pacifici, Nat. Photonics 10, 681 (2016).
[Crossref]

2015 (4)

2014 (5)

A. Norrman, T. Setälä, and A. T. Friberg, Opt. Express 22, 4628 (2014).
[Crossref]

A. Norrman, T. Setälä, and A. T. Friberg, Phys. Rev. A 90, 053849 (2014).
[Crossref]

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[Crossref]

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

L. Ma and S. A. Ponomarenko, Opt. Lett. 39, 6656 (2014).
[Crossref]

2013 (1)

2012 (2)

2009 (1)

R. W. Schoonover and T. D. Visser, Phys. Rev. A 79, 043809 (2009).
[Crossref]

2007 (1)

2005 (3)

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[Crossref]

E. Betzig, Opt. Express 13, 3021 (2005).
[Crossref]

S. A. Ponomarenko, H. Roychowdhury, and E. Wolf, Phys. Lett. A 345, 10 (2005).
[Crossref]

2003 (2)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, Nature 426, 421 (2003).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
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1994 (1)

P. Senthilkumaran and R. S. Sirohi, Opt. Commun. 105, 158 (1994).
[Crossref]

Aberra Guebrou, S.

Bellessa, J.

Bembenec, J. N.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Bessueille, F.

Betzig, E.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

E. Betzig, Opt. Express 13, 3021 (2005).
[Crossref]

Bloch, I.

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 1, 23 (2005).
[Crossref]

Böhme, R.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Cai, Y.

Y. Chen, S. A. Ponomarenko, and Y. Cai, Appl. Phys. Lett. 109, 061107 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, Opt. Lett. 41, 4182 (2016).
[Crossref]

Chen, B. C.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Chen, Y.

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, Opt. Lett. 43, 1395 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Phys. Rev. A 97, 041801 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Opt. Lett. 42, 3279 (2017).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, Appl. Phys. Lett. 109, 061107 (2016).
[Crossref]

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref]

Cui, K.

Dalibard, J.

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 1, 23 (2005).
[Crossref]

Davidson, M. W.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, Nature 426, 421 (2003).
[Crossref]

English, B. P.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Feng, X.

Friberg, A. T.

Fritz-Laylin, L.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Grill, S. W.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Hammer, J. A.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Huang, Y.

Janetopoulos, C.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Kiehart, D. P.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Laverdant, J.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref]

Legant, W. R.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Li, D.

D. Morrill, D. Li, and D. Pacifici, Nat. Photonics 10, 681 (2016).
[Crossref]

Lippincott-Schwartz, J.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Liu, F.

Liu, X.

Liu, Z.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Ma, L.

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, Nature 426, 421 (2003).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Mao, H.

Milkie, D. E.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Mimori-Kiyosue, Y.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Mitchell, D. M.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Morrill, D.

D. Morrill, D. Li, and D. Pacifici, Nat. Photonics 10, 681 (2016).
[Crossref]

Mullins, R. D.

B. C. Chen, W. R. Legant, K. Wang, L. Shao, D. E. Milkie, M. W. Davidson, C. Janetopoulos, X. S. Wu, J. A. Hammer, Z. Liu, B. P. English, Y. Mimori-Kiyosue, D. P. Romero, A. T. Ritter, J. Lippincott-Schwartz, L. Fritz-Laylin, R. D. Mullins, D. M. Mitchell, J. N. Bembenec, A. C. Reymann, R. Böhme, S. W. Grill, J. T. Wang, G. Seydoux, U. S. Tulu, D. P. Kiehart, and E. Betzig, Science 346, 1257998 (2014).
[Crossref]

Nascimbene, S.

I. Bloch, J. Dalibard, and S. Nascimbene, Nat. Phys. 1, 23 (2005).
[Crossref]

Norrman, A.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd ed. (Cambridge University, 2012).

Pacifici, D.

D. Morrill, D. Li, and D. Pacifici, Nat. Photonics 10, 681 (2016).
[Crossref]

Ponomarenko, S. A.

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Phys. Rev. A 97, 041801 (2018).
[Crossref]

H. Mao, Y. Chen, S. A. Ponomarenko, and A. T. Friberg, Opt. Lett. 43, 1395 (2018).
[Crossref]

Y. Chen, A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Opt. Lett. 42, 3279 (2017).
[Crossref]

A. Norrman, S. A. Ponomarenko, and A. T. Friberg, Europhys. Lett. 116, 64001 (2016).
[Crossref]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, Opt. Lett. 41, 4182 (2016).
[Crossref]

Y. Chen, S. A. Ponomarenko, and Y. Cai, Appl. Phys. Lett. 109, 061107 (2016).
[Crossref]

L. Ma and S. A. Ponomarenko, Opt. Express 23, 1848 (2015).
[Crossref]

L. Ma and S. A. Ponomarenko, Opt. Lett. 39, 6656 (2014).
[Crossref]

S. A. Ponomarenko, H. Roychowdhury, and E. Wolf, Phys. Lett. A 345, 10 (2005).
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Reymann, A. C.

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Figures (4)

Fig. 1.
Fig. 1. Schematic illustration of the synthesis of a partially coherent SPPL field. All SPP modes with their excitation points located uniformly on a circle of radius a at the metal–air interface (xy plane) propagate toward the coordinate origin O.
Fig. 2.
Fig. 2. Spatial behavior of the electromagnetic degree of coherence μ(r,ω) for uncorrelated SPPL fields at an Ag–air interface at the free-space wavelength λ=632.8  nm with (a) N=3, (b) N=5, (c) N=6, (d) N=8, (e) N=10, (f) N=12, (g) N=15, and (h) N=20 SPP modes. Empirical data are used for Ag [33].
Fig. 3.
Fig. 3. Spatial behavior of the degree of polarization P(r,ω) (top panels) and the azimuth angle ϕ(r,ω) (bottom panels) of the linear polarization state related to the tangential component of the uncorrelated SPPL field at an Ag–air interface at free-space wavelength λ=632.8  nm with N=3 (left column), N=5 (second column), N=6 (third column), and N=20 (right column) SPP modes. The red lines above the color bars in the bottom panels show the corresponding polarization orientations, and lSPP(λ) is the SPP propagation length. The relative permittivity of Ag is from empirical data [33].
Fig. 4.
Fig. 4. Spatial behavior of the spectral density S(r,ω) (top panels) and the ratio η(r,ω) (bottom panels) of the normal and tangential spectral density contributions to the correlated SPPL field at an Ag–air interface at the free-space wavelength λ=632.8  nm with N=3 (left column), N=5 (second column), N=6 (third column), and N=20 (right column) SPP modes. The excitation circle radius is equal to the SPP propagation length. Note that S(r,ω) in the top panels is normalized with respect to the initial SPP intensity ISPP(ω) and the mode number N squared. The over-exposed areas in the bottom panels stand for η(r,ω)>10. Empirical data [33] are used for the relative permittivity of Ag.

Equations (12)

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E(r,ω)=n=1NEn(ω)p^n(ω)eikn(ω)·(rr0n),
kn(ω)=kt(ω)e^n+kz(ω)e^z,
p^n(ω)=k^n(ω)×(e^z×e^n),
kt(ω)=ωc[εr(ω)εr(ω)+1]1/2,kz(ω)=ωc[1εr(ω)+1]1/2,
W(r1,r2,ω)=e2kt(ω)an=1Nm=1NWnm(ω)Knm(ω)×ei[km(ω)·r2kn*(ω)·r1],
W(r1,r2,ω)=ISPP(ω)e2kt(ω)an=1NKnn(ω)×ei[kn(ω)·r2kn*(ω)·r1].
μ(r1,r2,ω)=W(r1,r2,ω)F[trϕ(r1,ω)trϕ(r2,ω)]1/2,
μ2(r,ω)=1N+2n=1N1m=n+1Nξnm(r,ω)Nn=1Nξnn(r,ω),
ξnm(r,ω)=[1+κ2(ω)cosΔθnm1+κ2(ω)]2ekt(ω)(Cnm+x+Snm+y)×cos[kt(ω)(Cnmx+Snmy)]
ϕ(r,ω)=ISPP(ω)e2[kt(ω)a+kz(ω)z]n=1NKnn(ω)ξnn(r,ω),
ϕxx(r,ω)+ϕyy(r,ω)ϕzz(r,ω)=κ2(ω),
S(r,ω)=ISPP(ω)e2[kt(ω)a+kz(ω)z][n=1Nξnn(r,ω)+2n=1N1m=n+1N1+κ2(ω)1+κ2(ω)cosΔθnmξnm(r,ω)].