Abstract

As complex random electromagnetic fields are increasingly exploited in advanced photonics, the measurement of their statistical properties emerges as a crucial technical issue. For spatial coherence, we employ dipole scattering and report on a nano-optics counterpart of Young’s interferometer with the openings replaced by metallic dipolar nanoparticles. The results are in agreement with those obtained by other methods. The resolution is comparable to the size of the probes, well beyond that in standard techniques. While we consider here random optical beams, the two-probe method finds particular use in electromagnetic near-field optics where customary polarization elements cannot be employed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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    [Crossref]
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2016 (2)

2015 (2)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, Opt. Lett. 40, 2898 (2015).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

2014 (2)

2013 (1)

E. Massa, S. A. Maier, and V. Giannini, New J. Phys. 15, 063013 (2013).
[Crossref]

2011 (1)

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

2007 (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

2006 (3)

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

I. U. Vakarelski and K. Higashitani, Langmuir 22, 2931 (2006).
[Crossref]

M. Santarsiero and R. Borghi, Opt. Lett. 31, 861 (2006).
[Crossref]

2004 (1)

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

2003 (1)

2002 (1)

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

2001 (2)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

2000 (1)

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

1991 (1)

O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).
[Crossref]

1961 (1)

C. Jönsson, Z. Phys. 161, 454 (1961).
[Crossref]

1804 (1)

T. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[Crossref]

Banzer, P.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Bauer, T.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Borghi, R.

Boyd, R. W.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Braverman, B.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Brown, T.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

Carminati, R.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Carnal, O.

O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).
[Crossref]

Chen, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Dändliker, R.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

De Wilde, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Formanek, F.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Friberg, A. T.

Giannini, V.

E. Massa, S. A. Maier, and V. Giannini, New J. Phys. 15, 063013 (2013).
[Crossref]

Gralak, B.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Greffet, J.-J.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Grondalski, J.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Higashitani, K.

I. U. Vakarelski and K. Higashitani, Langmuir 22, 2931 (2006).
[Crossref]

Hoffmann, P.

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

James, D. F. V.

Jönsson, C.

C. Jönsson, Z. Phys. 161, 454 (1961).
[Crossref]

Joulain, K.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

Kalkbrenner, T.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

Karimi, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Kim, Y.-H.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Kocsis, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Kulik, S. P.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Lehtolahti, J.

Lemoine, P.-A.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Leppänen, L.-P.

Leuchs, G.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Li, D.

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

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

Maier, S. A.

E. Massa, S. A. Maier, and V. Giannini, New J. Phys. 15, 063013 (2013).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Marquis-Weible, F.

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

Marrucci, L.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Massa, E.

E. Massa, S. A. Maier, and V. Giannini, New J. Phys. 15, 063013 (2013).
[Crossref]

Mirin, R. P.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Mlynek, J.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).
[Crossref]

Morrill, D.

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

Mulet, J.-P.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

Nesci, A.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Orlov, S.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Pacifici, D.

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

Partanen, H.

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

Ramstein, M.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

Ravets, S.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Rubano, A.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Saastamoinen, K.

Sandoghdar, V.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

Santamato, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Santarsiero, M.

Scully, M. O.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Setälä, T.

Shalm, L. K.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

Shih, Y.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

Sqalli, O.

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

Steinberg, A. M.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Stevens, M. J.

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

Tervo, J.

Tortora, P.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

Turunen, J.

Utke, I.

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

Vaccaro, L.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

Vakarelski, I. U.

I. U. Vakarelski and K. Higashitani, Langmuir 22, 2931 (2006).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Young, T.

T. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[Crossref]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

Yu, R.

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

J. Appl. Phys. (1)

O. Sqalli, I. Utke, P. Hoffmann, and F. Marquis-Weible, J. Appl. Phys. 92, 1078 (2002).
[Crossref]

J. Microsc. (1)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, J. Microsc. 202, 72 (2001).
[Crossref]

J. Opt. A (1)

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, J. Opt. A 6, S18 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

Langmuir (1)

I. U. Vakarelski and K. Higashitani, Langmuir 22, 2931 (2006).
[Crossref]

Nat. Photonics (2)

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

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, Nat. Photonics 1, 228 (2007).
[Crossref]

Nature (1)

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, Nature 444, 740 (2006).
[Crossref]

New J. Phys. (1)

E. Massa, S. A. Maier, and V. Giannini, New J. Phys. 15, 063013 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Philos. Trans. R. Soc. London (1)

T. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[Crossref]

Phys. Rev. Lett. (3)

O. Carnal and J. Mlynek, Phys. Rev. Lett. 66, 2689 (1991).
[Crossref]

Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[Crossref]

Science (2)

S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011).
[Crossref]

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, Science 347, 964 (2015).
[Crossref]

Z. Phys. (1)

C. Jönsson, Z. Phys. 161, 454 (1961).
[Crossref]

Other (3)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Supplementary Material (1)

NameDescription
» Supplement 1       Supplemental document on used methods, analysis and fabrication.

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Figures (3)

Fig. 1.
Fig. 1. Two-probe measurement setup. (a) Top view of the geometry. Light from a single-mode or multimode He–Ne laser is expanded, collimated, and polarized vertically. The beam is focused on a pair of gold (Au) nanocubes that act as dipolar scatterers. The far-field interference pattern produced by the particles is recorded at an angle θ to the substrate by moving a photodetector in the vertical direction (parallel to polarization). (b) Intensity distribution across the expanded, unfocused multimode beam. (c) Cross sections of the intensity distribution (red dashed line) and the degree of coherence (blue line) along the yellow line in (b). lc=84  μm marks the transverse coherence length. (d) Scattering cross section σ of cubic Au particles in terms of the side length L and wavelength λ. The chosen particle size, L=130  nm, and wavelength, λ=632.8  nm, are marked with blue lines. (e) SEM image of an actual Au nanocube pair placed on silicon (Si) substrate, with a magnified image of a single cube.
Fig. 2.
Fig. 2. Illustration of the far-field intensity patterns averaged over several measurements (black dots) and fitted curves (blue solid lines) as a function of the detector position s. (a) Scaled-down (focused) single-mode laser beam when the particles are 3 μm apart. (b)–(e) Scaled multimode laser beam when the particles are separated by a distance of a=3, 4, 5, and 7 μm, respectively.
Fig. 3.
Fig. 3. Measured (average) degrees of spatial coherence for the multimode laser beam. The colored geometrical objects represent the various particle separations a. The solid blue curve shows the degree of coherence of the input beam obtained by a DMD. The resulting degree of coherence profile is scaled down by a factor of 20 to account for the focusing of the beam. The red cross corresponds to the single-mode laser with a 3 μm particle separation, but is here associated with the multimode laser with a vanishingly small particle distance. The errors (black bars) are estimated by computing the standard deviation of multiple measurements.

Equations (2)

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V=ImaxIminImax+Imin=2I1I2I1+I2|μ(r1,r2,ω)|,
|μ(r1,r2,ω)|=V,

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