Abstract

Surface plasmon resonance is conventionally conducted in the visible range and, during the past decades, it has proved its efficiency in probing molecular scale interactions. Here we elaborate on the first implementation of a high resolution surface plasmon microscope that operates at near infrared (IR) wavelength for the specific purpose of living matter imaging. We analyze the characteristic angular and spatial frequencies of plasmon resonance in visible and near IR lights and how these combined quantities contribute to the V (Z) response of a scanning surface plasmon microscope (SSPM). Using a space-frequency wavelet decomposition, we show that the V (Z) response of the SSPM for red (632.8 nm) and near IR (1550 nm) lights includes the frequential response of plasmon resonance together with additional parasitic frequencies induced by the objective pupil. Because the objective lens pupil profile is often unknown, this space-frequency decomposition turns out to be very useful to decipher the characteristic frequencies of the experimental V (Z) curves. Comparing the visible and near IR light responses of the SSPM, we show that our objective lens, primarily designed for visible light microscopy, is still operating very efficiently in near IR light. Actually, despite their loss in resolution, the SSPM images obtained with near IR light remain contrasted for a wider range of defocus values from negative to positive Z values. We illustrate our theoretical modeling with a preliminary experimental application to blood cell imaging.

© 2013 OSA

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  64. R. Vander and S. G. Lipson, “High-resolution surface-plasmon resonance real-time imaging,” Opt. Lett.34(1), 37–39 (2009).
    [CrossRef]

2012 (2)

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
[CrossRef]

B. Zhang, S. Pechprasarn, J. Zhang, and M. G. Somekh, “Confocal surface plasmon microscopy with pupil function engineering.” Opt. Express20(7), 7388–97 (2012).
[CrossRef] [PubMed]

2011 (2)

A. Arneodo, C. Vaillant, B. Audit, F. Argoul, Y. DAubenton-Carafa, and C. Thermes, “Multi-scale coding of genomic information: From DNA sequence to genome structure and function,” Physics Reports498(2–3), 45–188 (2011).
[CrossRef]

L. Berguiga, T. Roland, K. Monier, J. Elezgaray, and F. Argoul, “Amplitude and phase images of cellular structures with a scanning surface plasmon microscope,” Opt. Express19(7), 2829–2836 (2011).
[CrossRef]

2010 (4)

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter81(23), 235,419 (2010).
[CrossRef]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A27(3), 450–457 (2010).
[CrossRef]

J. Elezgaray, L. Berguiga, and F. Argoul, “Optimization of branched resonant nanostructures illuminated by a strongly focused beam,” Appl. Phys. Lett.97(24), 243,103 (2010).
[CrossRef]

F. Argoul, K. Monier, T. Roland, J. Elezgaray, and L. Berguiga, “High resolution surface plasmon microscopy for cell imaging,” in SPIE Proc. Biophotonics: Photonic Solutions for Better Health Care II, p. 771506 (2010).
[CrossRef]

2009 (10)

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
[CrossRef]

S. Patskovsky, M. Vallieres, M. Maisonneuve, I.-H. Song, M. Meunier, and A. V. Kabashin, “Designing efficient zero calibration point for phase-sensitive surface plasmon resonance biosensing,” Opt. Express17(4), 2255–2263 (2009).
[CrossRef] [PubMed]

R. Jha and A. K. Sharma, “High-performance sensor based on surface plasmon resonance with chalcogenide prism and aluminum for detection in infrared,” Opt. Lett.34(6), 749–751 (2009).
[CrossRef] [PubMed]

R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt.11(4), 045,502 (2009).
[CrossRef]

M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, “Midinfrared surface-plasmon resonance: A novel biophysical tool for studying living cells,” J. Appl. Phys.105(10), 102,036 (2009).
[CrossRef]

V. Yashunsky, S. Shimron, V. Lirtsman, A. M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, and B. Aroeti, “Real-time monitoring of transferrin-induced endocytic vesicle formation by mid-infrared surface plasmon resonance.” Biophys. J.97(4), 1003–1012 (2009).
[CrossRef] [PubMed]

M. G. Somekh, G. Stabler, S. Liu, J. Zhang, and C. W. See, “Wide field high resolution surface plasmon interference microscopy,” Opt. Lett.34(20), 3110–3112 (2009).
[CrossRef] [PubMed]

K. Watanabe, G. Terakado, and H. Kano, “Localized surface plasmon microscope with an illumination system employing a radially polarized zeroth-order Bessel beam,” Opt. Lett.34(8), 1180–1182 (2009).
[CrossRef] [PubMed]

R. Vander and S. G. Lipson, “High-resolution surface-plasmon resonance real-time imaging,” Opt. Lett.34(1), 37–39 (2009).
[CrossRef]

2008 (2)

A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia3, 4103 (2008).
[CrossRef]

J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

2007 (1)

2005 (2)

J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt.7(9), 472–478 (2005).
[CrossRef]

V. Lirtsman, R. Ziblat, M. Golosovsky, D. Davidov, R. Pogreb, V. Sacks-Granek, and J. Rishpon, “Surface-plasmon resonance with infrared excitation: Studies of phospholipid membrane growth,” J. Appl. Phys.98(9), 093,506 (2005).
[CrossRef]

2004 (3)

G. Stabler, M. G. Somekh, and C. W. See, “High-resolution wide-field surface plasmon microscopy,” J. Microsc.214(Pt 3), 328–333 (2004).
[CrossRef]

J. Zhang, C. W. See, M. G. Somekh, M. C. Pitter, and S. G. Liu, “Wide-field surface plasmon microscopy with solid immersion excitation,” Appl. Phys. Lett.85(22), 5451–5453 (2004).
[CrossRef]

P. Kestener and A. Arneodo, “Generalizing the wavelet-based multifractal formalism to random vector fields: Application to three-dimensional turbulence velocity and vorticity data,” Phys. Rev. Lett.93(4), 044,501 (2004).
[CrossRef]

2003 (3)

A. Arneodo, N. Decoster, P. Kesterner, and S. G. Roux, “A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications,” Advanced in Imaging and Electron Physics126, 1–92 (2003).
[CrossRef]

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem.377(3), 528–539 (2003).
[CrossRef] [PubMed]

S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. T. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt.42(34), 6905–6909 (2003).
[CrossRef] [PubMed]

2001 (1)

B. P. Nelson, T. E. Grimsrud, M. R. Liles, R. M. Goodman, and R. M. Corn, “Surface plasmon resonance imaging measurements of DNA and RNA hybridization adsorption onto DNA microarrays,” Anal. Chem.73(1), 1–7 (2001).
[CrossRef] [PubMed]

2000 (5)

P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk, O. A. Savchuk, G. Steiner, C. Kuhne, A. Huebner, and R. Salzer, “Surface plasmon resonance interferometry for micro-array biosensing,” Sens. Actuators, A85(1–3), 189–193 (2000).

A. G. Notcovich, V. Zhuk, and S. G. Lipson, “Surface plasmon resonance phase imaging,” Appl. Phys. Lett.76(13), 1665–1667 (2000).
[CrossRef]

A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun.174(January), 151–155 (2000).
[CrossRef]

M. G. Somekh, S. G. Liu, T. S. Velinov, and C. W. See, “Optical V(z) for high-resolution 2π surface plasmon microscopy,” Opt. Lett.25(11), 823–825 (2000).
[CrossRef]

H. Kano and W. Knoll, “A scanning microscope employing localized surface-plasmon-polaritons as a sensing probe,” Opt. Commun.182(August), 11–15 (2000).
[CrossRef]

1998 (1)

H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. A15(4), 1381–1386 (1998).
[CrossRef]

1995 (1)

A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A213(1–2), 232–275 (1995).
[CrossRef]

1992 (1)

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, and A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A: At. Mol. Opt. Phys.45(12), 8961–8964 (1992).
[CrossRef]

1990 (1)

E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett.64(7), 745–748 (1990).
[CrossRef] [PubMed]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B: Condens. Matter33(8), 5186–5201 (1986).
[CrossRef]

1984 (3)

C. Ilett, M. Somekh, and G. Briggs, “Acoustic microscopy of elastic discontinuities,” Proc. Roy. Soc. Lond. Ser. A393, 171–183 (1984).
[CrossRef]

P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration23, 85–102 (1984).
[CrossRef]

A. Grossmann and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal.15(4), 723–736 (1984).
[CrossRef]

1982 (2)

J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics47(2), 203–221 (1982).
[CrossRef]

J. Morlet, G. Arensz, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part II: Sampling theory and complex waves,” Geophysics41(2), 222–236 (1982).
[CrossRef]

1979 (1)

A. Atalar, “A physical model for acoustic signatures,” J. Appl. Phys.50, 8237–8239 (1979).
[CrossRef]

1978 (1)

A. Atalar, “An angular-spectrum approach to contrast in reflection acoustic microscopy,” J. Appl. Phys.40, 5130–5139 (1978).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B: Condens. Matter6(12), 4370–4379 (1972).
[CrossRef]

1968 (2)

A. Otto, “Excitation of nonradiative surface plasmon waves in silver by the method of frustrated total reflection,” Z. Angew. Phys.410, 398–410 (1968).

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excitated by light,” Z. Natur-forsch., A: Phys. Sci.23, 2135 (1968).

Adam, J.-L.

J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

Ali, S. T.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, UK, 2008).

Anne, M.-L.

J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

Antoine, J. P.

J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, UK, 2008).

Arenss, G.

J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics47(2), 203–221 (1982).
[CrossRef]

Arensz, G.

J. Morlet, G. Arensz, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part II: Sampling theory and complex waves,” Geophysics41(2), 222–236 (1982).
[CrossRef]

Argoul, F.

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
[CrossRef]

L. Berguiga, T. Roland, K. Monier, J. Elezgaray, and F. Argoul, “Amplitude and phase images of cellular structures with a scanning surface plasmon microscope,” Opt. Express19(7), 2829–2836 (2011).
[CrossRef]

A. Arneodo, C. Vaillant, B. Audit, F. Argoul, Y. DAubenton-Carafa, and C. Thermes, “Multi-scale coding of genomic information: From DNA sequence to genome structure and function,” Physics Reports498(2–3), 45–188 (2011).
[CrossRef]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A27(3), 450–457 (2010).
[CrossRef]

J. Elezgaray, L. Berguiga, and F. Argoul, “Optimization of branched resonant nanostructures illuminated by a strongly focused beam,” Appl. Phys. Lett.97(24), 243,103 (2010).
[CrossRef]

F. Argoul, K. Monier, T. Roland, J. Elezgaray, and L. Berguiga, “High resolution surface plasmon microscopy for cell imaging,” in SPIE Proc. Biophotonics: Photonic Solutions for Better Health Care II, p. 771506 (2010).
[CrossRef]

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter81(23), 235,419 (2010).
[CrossRef]

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
[CrossRef]

L. Berguiga, S. Zhang, F. Argoul, and J. Elezgaray, “High-resolution surface-plasmon imaging in air and in water: V(z) curve and operating conditions,” Opt. Lett.32(5), 509–511 (2007).
[CrossRef] [PubMed]

J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt.7(9), 472–478 (2005).
[CrossRef]

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, and A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A: At. Mol. Opt. Phys.45(12), 8961–8964 (1992).
[CrossRef]

E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett.64(7), 745–748 (1990).
[CrossRef] [PubMed]

L. Berguiga, E. Boyer-Provera, J. Elezgaray, and F. Argoul, “Sensing nanometer depth of focused optical fields with scanning surface plasmon microscopy,” Plasmonics, in press (2012).
[CrossRef]

A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences: de l’ADN aux croissances cristallines (Diderot Editeur, Art et Sciences, Paris, 1995).

Arneodo, A.

A. Arneodo, C. Vaillant, B. Audit, F. Argoul, Y. DAubenton-Carafa, and C. Thermes, “Multi-scale coding of genomic information: From DNA sequence to genome structure and function,” Physics Reports498(2–3), 45–188 (2011).
[CrossRef]

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia3, 4103 (2008).
[CrossRef]

P. Kestener and A. Arneodo, “Generalizing the wavelet-based multifractal formalism to random vector fields: Application to three-dimensional turbulence velocity and vorticity data,” Phys. Rev. Lett.93(4), 044,501 (2004).
[CrossRef]

A. Arneodo, N. Decoster, P. Kesterner, and S. G. Roux, “A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications,” Advanced in Imaging and Electron Physics126, 1–92 (2003).
[CrossRef]

A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A213(1–2), 232–275 (1995).
[CrossRef]

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, and A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A: At. Mol. Opt. Phys.45(12), 8961–8964 (1992).
[CrossRef]

E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett.64(7), 745–748 (1990).
[CrossRef] [PubMed]

A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences: de l’ADN aux croissances cristallines (Diderot Editeur, Art et Sciences, Paris, 1995).

Aroeti, B.

M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, “Midinfrared surface-plasmon resonance: A novel biophysical tool for studying living cells,” J. Appl. Phys.105(10), 102,036 (2009).
[CrossRef]

V. Yashunsky, S. Shimron, V. Lirtsman, A. M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, and B. Aroeti, “Real-time monitoring of transferrin-induced endocytic vesicle formation by mid-infrared surface plasmon resonance.” Biophys. J.97(4), 1003–1012 (2009).
[CrossRef] [PubMed]

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A. Arneodo, C. Vaillant, B. Audit, F. Argoul, Y. DAubenton-Carafa, and C. Thermes, “Multi-scale coding of genomic information: From DNA sequence to genome structure and function,” Physics Reports498(2–3), 45–188 (2011).
[CrossRef]

A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia3, 4103 (2008).
[CrossRef]

Bacry, E.

A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A213(1–2), 232–275 (1995).
[CrossRef]

A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences: de l’ADN aux croissances cristallines (Diderot Editeur, Art et Sciences, Paris, 1995).

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P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk, O. A. Savchuk, G. Steiner, C. Kuhne, A. Huebner, and R. Salzer, “Surface plasmon resonance interferometry for micro-array biosensing,” Sens. Actuators, A85(1–3), 189–193 (2000).

A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun.174(January), 151–155 (2000).
[CrossRef]

Berguiga, L.

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
[CrossRef]

L. Berguiga, T. Roland, K. Monier, J. Elezgaray, and F. Argoul, “Amplitude and phase images of cellular structures with a scanning surface plasmon microscope,” Opt. Express19(7), 2829–2836 (2011).
[CrossRef]

F. Argoul, K. Monier, T. Roland, J. Elezgaray, and L. Berguiga, “High resolution surface plasmon microscopy for cell imaging,” in SPIE Proc. Biophotonics: Photonic Solutions for Better Health Care II, p. 771506 (2010).
[CrossRef]

J. Elezgaray, L. Berguiga, and F. Argoul, “Optimization of branched resonant nanostructures illuminated by a strongly focused beam,” Appl. Phys. Lett.97(24), 243,103 (2010).
[CrossRef]

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter81(23), 235,419 (2010).
[CrossRef]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A27(3), 450–457 (2010).
[CrossRef]

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
[CrossRef]

L. Berguiga, S. Zhang, F. Argoul, and J. Elezgaray, “High-resolution surface-plasmon imaging in air and in water: V(z) curve and operating conditions,” Opt. Lett.32(5), 509–511 (2007).
[CrossRef] [PubMed]

L. Berguiga, E. Boyer-Provera, J. Elezgaray, and F. Argoul, “Sensing nanometer depth of focused optical fields with scanning surface plasmon microscopy,” Plasmonics, in press (2012).
[CrossRef]

Bonneviot, L.

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
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M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 7th edition, 1999).

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J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

Boyer-Provera, E.

L. Berguiga, E. Boyer-Provera, J. Elezgaray, and F. Argoul, “Sensing nanometer depth of focused optical fields with scanning surface plasmon microscopy,” Plasmonics, in press (2012).
[CrossRef]

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C. Ilett, M. Somekh, and G. Briggs, “Acoustic microscopy of elastic discontinuities,” Proc. Roy. Soc. Lond. Ser. A393, 171–183 (1984).
[CrossRef]

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J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
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P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B: Condens. Matter6(12), 4370–4379 (1972).
[CrossRef]

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J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

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J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

Corn, R. M.

B. P. Nelson, T. E. Grimsrud, M. R. Liles, R. M. Goodman, and R. M. Corn, “Surface plasmon resonance imaging measurements of DNA and RNA hybridization adsorption onto DNA microarrays,” Anal. Chem.73(1), 1–7 (2001).
[CrossRef] [PubMed]

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DAubenton-Carafa, Y.

A. Arneodo, C. Vaillant, B. Audit, F. Argoul, Y. DAubenton-Carafa, and C. Thermes, “Multi-scale coding of genomic information: From DNA sequence to genome structure and function,” Physics Reports498(2–3), 45–188 (2011).
[CrossRef]

Davidov, D.

V. Yashunsky, S. Shimron, V. Lirtsman, A. M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, and B. Aroeti, “Real-time monitoring of transferrin-induced endocytic vesicle formation by mid-infrared surface plasmon resonance.” Biophys. J.97(4), 1003–1012 (2009).
[CrossRef] [PubMed]

M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, “Midinfrared surface-plasmon resonance: A novel biophysical tool for studying living cells,” J. Appl. Phys.105(10), 102,036 (2009).
[CrossRef]

V. Lirtsman, R. Ziblat, M. Golosovsky, D. Davidov, R. Pogreb, V. Sacks-Granek, and J. Rishpon, “Surface-plasmon resonance with infrared excitation: Studies of phospholipid membrane growth,” J. Appl. Phys.98(9), 093,506 (2005).
[CrossRef]

Decoster, N.

A. Arneodo, N. Decoster, P. Kesterner, and S. G. Roux, “A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications,” Advanced in Imaging and Electron Physics126, 1–92 (2003).
[CrossRef]

Delichère, P.

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

Deputier, S.

J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
[CrossRef]

Elezgaray, J.

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
[CrossRef]

L. Berguiga, T. Roland, K. Monier, J. Elezgaray, and F. Argoul, “Amplitude and phase images of cellular structures with a scanning surface plasmon microscope,” Opt. Express19(7), 2829–2836 (2011).
[CrossRef]

J. Elezgaray, L. Berguiga, and F. Argoul, “Optimization of branched resonant nanostructures illuminated by a strongly focused beam,” Appl. Phys. Lett.97(24), 243,103 (2010).
[CrossRef]

F. Argoul, K. Monier, T. Roland, J. Elezgaray, and L. Berguiga, “High resolution surface plasmon microscopy for cell imaging,” in SPIE Proc. Biophotonics: Photonic Solutions for Better Health Care II, p. 771506 (2010).
[CrossRef]

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter81(23), 235,419 (2010).
[CrossRef]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A27(3), 450–457 (2010).
[CrossRef]

T. Roland, A. Khalil, A. Tanenbaum, L. Berguiga, P. Delichère, L. Bonneviot, J. Elezgaray, A. Arneodo, and F. Argoul, “Revisiting the physical processes of vapodeposited thin gold films on chemically modified glass by atomic force and surface plasmon microscopies,” Surf. Sci.603(22), 3307–3320 (2009).
[CrossRef]

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
[CrossRef]

L. Berguiga, S. Zhang, F. Argoul, and J. Elezgaray, “High-resolution surface-plasmon imaging in air and in water: V(z) curve and operating conditions,” Opt. Lett.32(5), 509–511 (2007).
[CrossRef] [PubMed]

J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt.7(9), 472–478 (2005).
[CrossRef]

L. Berguiga, E. Boyer-Provera, J. Elezgaray, and F. Argoul, “Sensing nanometer depth of focused optical fields with scanning surface plasmon microscopy,” Plasmonics, in press (2012).
[CrossRef]

A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy, Ondelettes, Multifractales et Turbulences: de l’ADN aux croissances cristallines (Diderot Editeur, Art et Sciences, Paris, 1995).

Fahys, A.

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
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P. Flandrin, Temps-Fréquence (Hermès, France1993).

Fourgeau, E.

J. Morlet, G. Arensz, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part II: Sampling theory and complex waves,” Geophysics41(2), 222–236 (1982).
[CrossRef]

J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics47(2), 203–221 (1982).
[CrossRef]

Freysz, E.

J. F. Muzy, B. Pouligny, E. Freysz, F. Argoul, and A. Arneodo, “Optical-diffraction measurement of fractal dimensions and f(α) spectrum,” Phys. Rev. A: At. Mol. Opt. Phys.45(12), 8961–8964 (1992).
[CrossRef]

E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett.64(7), 745–748 (1990).
[CrossRef] [PubMed]

Giard, D.

J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics47(2), 203–221 (1982).
[CrossRef]

J. Morlet, G. Arensz, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part II: Sampling theory and complex waves,” Geophysics41(2), 222–236 (1982).
[CrossRef]

Golosovsky, M.

V. Yashunsky, S. Shimron, V. Lirtsman, A. M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, and B. Aroeti, “Real-time monitoring of transferrin-induced endocytic vesicle formation by mid-infrared surface plasmon resonance.” Biophys. J.97(4), 1003–1012 (2009).
[CrossRef] [PubMed]

M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, “Midinfrared surface-plasmon resonance: A novel biophysical tool for studying living cells,” J. Appl. Phys.105(10), 102,036 (2009).
[CrossRef]

V. Lirtsman, R. Ziblat, M. Golosovsky, D. Davidov, R. Pogreb, V. Sacks-Granek, and J. Rishpon, “Surface-plasmon resonance with infrared excitation: Studies of phospholipid membrane growth,” J. Appl. Phys.98(9), 093,506 (2005).
[CrossRef]

Goodman, R. M.

B. P. Nelson, T. E. Grimsrud, M. R. Liles, R. M. Goodman, and R. M. Corn, “Surface plasmon resonance imaging measurements of DNA and RNA hybridization adsorption onto DNA microarrays,” Anal. Chem.73(1), 1–7 (2001).
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P. I. Nikitin, A. N. Grigorenko, A. A. Beloglazov, M. V. Valeiko, A. I. Savchuk, O. A. Savchuk, G. Steiner, C. Kuhne, A. Huebner, and R. Salzer, “Surface plasmon resonance interferometry for micro-array biosensing,” Sens. Actuators, A85(1–3), 189–193 (2000).

A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun.174(January), 151–155 (2000).
[CrossRef]

Grimsrud, T. E.

B. P. Nelson, T. E. Grimsrud, M. R. Liles, R. M. Goodman, and R. M. Corn, “Surface plasmon resonance imaging measurements of DNA and RNA hybridization adsorption onto DNA microarrays,” Anal. Chem.73(1), 1–7 (2001).
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P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration23, 85–102 (1984).
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J. Le Person, F. Colas, C. Compère, M. Lehaitre, M.-L. Anne, C. Boussard-Plédel, B. Bureau, J.-L. Adam, S. Deputier, and M. Guilloux-Viry, “Surface plasmon resonance in chalcogenide glass-based optical system,” Sens. Actuators, B130(2), 771–776 (2008).
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Hugo, N.

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
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C. Ilett, M. Somekh, and G. Briggs, “Acoustic microscopy of elastic discontinuities,” Proc. Roy. Soc. Lond. Ser. A393, 171–183 (1984).
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A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia3, 4103 (2008).
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Advanced in Imaging and Electron Physics (1)

A. Arneodo, N. Decoster, P. Kesterner, and S. G. Roux, “A wavelet-based method for multifractal image analysis: from theoretical concepts to experimental applications,” Advanced in Imaging and Electron Physics126, 1–92 (2003).
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Appl. Opt. (1)

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J. Zhang, C. W. See, M. G. Somekh, M. C. Pitter, and S. G. Liu, “Wide-field surface plasmon microscopy with solid immersion excitation,” Appl. Phys. Lett.85(22), 5451–5453 (2004).
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A. G. Notcovich, V. Zhuk, and S. G. Lipson, “Surface plasmon resonance phase imaging,” Appl. Phys. Lett.76(13), 1665–1667 (2000).
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Biophys. J. (1)

V. Yashunsky, S. Shimron, V. Lirtsman, A. M. Weiss, N. Melamed-Book, M. Golosovsky, D. Davidov, and B. Aroeti, “Real-time monitoring of transferrin-induced endocytic vesicle formation by mid-infrared surface plasmon resonance.” Biophys. J.97(4), 1003–1012 (2009).
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C. R. Phys. (1)

F. Argoul, T. Roland, A. Fahys, L. Berguiga, and J. Elezgaray, “Uncovering phase maps from surface plasmon resonance images: Towards a sub-wavelength resolution,” C. R. Phys.13(8), 800–814 (2012).
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Frontiers of Physics in China (1)

S. Zhang, L. Berguiga, J. Elezgaray, N. Hugo, W. Li, T. Roland, H. Zeng, and F. Argoul, “Advances in surface plasmon resonance-based high throughput biochips,” Frontiers of Physics in China4(4), 469–480 (2009).
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J. Appl. Phys. (4)

M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, “Midinfrared surface-plasmon resonance: A novel biophysical tool for studying living cells,” J. Appl. Phys.105(10), 102,036 (2009).
[CrossRef]

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R. Jha and A. K. Sharma, “Chalcogenide glass prism based SPR sensor with Ag-Au bimetallic nanoparticle alloy in infrared wavelength region,” J. Opt. A: Pure Appl. Opt.11(4), 045,502 (2009).
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J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A27(3), 450–457 (2010).
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Opt. Commun. (2)

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Figures (13)

Fig. 1
Fig. 1

(a) Global view of the optical set-up. (b) Zoom on the inverted microscope stand and the 3 axis piezo-table (green arrows) with the gold coated coverslip. S1 and S2 are respectively the red (He-Ne - 632.8 nm) and the near-IR (1550 nm) sources. BS1 and BS2 are the two beam splitters. M1 and M2 are the two mirrors of the two interferometers. AOs are acousto-optic modulators; each interferometer has two AOs, designed for each wavelength. On each interferometer, before the injection in the microscope stand, we have interposed beam expanders (BE). Finally the two red (1) and near-IR (2) beams are injected inside a microscope stand (MS) toward the microscope objective lens (OL) which focuses onto the sample coverslip (CV). D1 (resp. D2) is an amplified silicon (resp. germanium) detector.

Fig. 2
Fig. 2

Plot of the evolution of LX (a) and LZSP (b) with the wavelength λ. The dielectric permittivities of gold εm(λ) were taken from reference [35].

Fig. 3
Fig. 3

Reflectivity curves (Rp) computed for a 35 nm gold film with a 3 nm chromium sublayer, with air as dielectric medium. (a) Modulus of Rp versus θ. (b) 3D plots of the real and imaginary parts of Rp and the angle of incidence θ: [ℜ(Rp), ℑ(Rp), θ]. In red (resp. black) are plotted the curves for λ = 632.8 nm (resp. λ = 1550 nm). The dielectric permittivities εm(λ) of gold were taken from reference [35], and the ones of glass are those of BK7.

Fig. 4
Fig. 4

Plot of the evolution of θSP (a) and ΔθSP (b) with the wavelength λ, computed for a 35 nm gold film with a 3 nm chromium sublayer, with air as dielectric medium. The red curve in (a) corresponds to the total internal reflection angle θATR. The dielectric permittivities εm(λ) of gold were taken from reference [35], and the ones of glass are those of BK7.

Fig. 5
Fig. 5

Color maps of the reflectivity modulus (|Rp|) in the two dimensional space [θ, e], for e varying from 0 to 250 nm. |Rp| is color-coded from dark blue (0) to carmine red (1). (a) λ = 632.8 nm; (b) λ = 1550 nm. As explained in the text, we took a thin 35 nm gold film on a 3 nm sublayer of chromium, and a dielectric layer with thickness e in contact with air.

Fig. 6
Fig. 6

3D plots of the real and imaginary parts of Rp and the angle of incidence θ: [ℜ(Rp), ℑ(Rp), θ]. The differents curves were computed for a 35 nm gold film with a 3 nm sublayer of chromium and a dielectric layer (index 1.4) of increasing thickness from 0 nm (red) to 250 nm (blue). (a) λ = 632.8 nm; (b) λ = 1550 nm.

Fig. 7
Fig. 7

Plot of the experimental |V (Z)| curves obtained with the two-wavelength SSPM set-up. (a) Lin-lin plots of the normalized modulus responses |V (Z)|/V0. (b) Log-lin plot of |V (Z)|. In red: visible light (λ = 632.8 nm); in black: near IR light (λ = 1550 nm).

Fig. 8
Fig. 8

Modeling the objective lens pupil from a rectangular shape (P = 1) to a smooth profile. The σ parameter describes the smoothing of P (Eq. (10)). The dashed black (resp. solid red) vertical lines correspond to the resonance plasmon angles θSP for near IR (resp. red) light.

Fig. 9
Fig. 9

Theoretical |V (Z)| curves (in log-lin representation) computed for a constant reflectivity Rp = 1 (panels (a) and (b)) and for a 35 nm gold film with a 3 nm sublayer of chromium in air (panels (c) and (d)). The P shape parameter σ (from 0 to 1) accounts for the smoothing of the objective lens pupil (Fig. 8). (a, c) λ = 1550 nm. (b, d) λ = 632.8 nm.

Fig. 10
Fig. 10

Morlet wavelet transform of the V (Z) functions computed for a 35 nm gold film with a 3 nm chromium sublayer, for visible light (λ = 632.8 nm). (Bottom) ln|V (Z)| vs Z; (Top) |Tln|V|(Z,ν = 1/a)| as coded using 256 colors from blue (minimal value) to red (maximal value). (a) Rectangular pupil function P (σ = 0). (b) Smoothed pupil function P (σ = 0.17). (c) Smoothed pupil function P (σ = 0.81). θmax = arcsin(NA/nc) with NA= 1.45 and nc = 1.5151.

Fig. 11
Fig. 11

Morlet wavelet transform of the V (Z) functions computed for a 35 nm gold film with a 3 nm chromium sublayer, for near IR light (λ = 1550 nm). Same representation as in Fig. 10. (a) Rectangular pupil function P (σ = 0). (b) Smoothed pupil function P (σ = 0.17). (c) Smoothed pupil function P (σ = 0.81). θmax = arcsin(NA/nc) with NA= 1.45 and nc = 1.5.

Fig. 12
Fig. 12

Morlet wavelet transform of the experimental V (Z) functions obtained with the SSPM optical set-up at (a) visible (λ = 632.8 nm) and (b) near IR (λ = 1550 nm) wavelengths. These V (Z) curves were recorded for a 35 nm thick gold film with a 3 nm sublayer of chromium in air. Same representation as in Fig. 10

Fig. 13
Fig. 13

Comparison of SSPM images collected from a fixed lymphocyte in air with (a–d) visible light (λ = 632.8 nm) and (a’–d’) IR light (λ = 1550 nm). (a) Z = −1.7μm, (b) Z = −0.2μm, (c) Z = 1.25μm, (d) Z = 3.95μm. (a’) Z = −2μm, (b’) Z = −0.5μm, (c’) Z = 1.3μm, (d’) Z = 4.35μm. The scale bar is 10 μm.

Tables (2)

Tables Icon

Table 1 Table of spatial frequencies of V (Z) computed from a constant pupil function and a numerical aperture lens NA=1.45

Tables Icon

Table 2 Table of spatial frequencies (Eq. (23)) corresponding to the combination of plasmon resonance and objective pupil lens aperture, computed for a 35 nm gold film with a 3 nm sublayer of chromium.

Equations (30)

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k S P = 2 π λ ε D ε m ε D + ε m + Δ k ,
k E W = 2 π λ n c sin θ = k S P = k S P + i k S P ,
L X = 1 2 k S P = λ 2 π { ε m + ε D ε m ε D } 3 / 2 ε m 2 ε m .
l Z S P = λ 2 π { | ε m + ε D | ε D 2 } 1 / 2 ,
θ S P = arcsin { k S P k c } ,
Δ θ S P = k S P k c 2 k S P 2 .
V ( Z ) = θ min θ max P 2 ( θ ) 𝒬 ( θ ) e 2 j ( 2 π n c / λ ) Z cos θ sin θ d θ ,
𝒬 ( θ ) = 0 2 π R p ( sin θ ) cos 2 φ d φ = π R p ( sin θ ) ;
𝒬 ( θ ) = 0 2 π R s ( sin θ ) sin 2 φ d φ = π R s ( sin θ ) .
P ( cos θ ) = [ 1 exp ( cos θ 2 2 σ 2 ) ] .
ν = ( 2 n c / λ ) cos θ , d ν = ( 2 n c / λ ) sin θ d θ ,
V ( Z ) = λ 2 n c ν min ν max P 2 ( ν ) 𝒬 ( ν ) e 2 π j ν Z d ν ,
ν min = ( 2 n c / λ ) cos θ max and ν max = ( 2 n c / λ ) cos θ min .
V ( Z ) = λ 2 n c P ¯ 2 ( ν ) 𝒬 ( ν ) e 2 π j ν Z d ν .
V ( Z ) = λ 2 n c P ¯ 2 ( ν ) e 2 π j ν Z d ν ,
V ( Z ) = λ 2 n c 2 n c λ cos θ max 2 n c λ cos θ min e 2 π j ν Z d ν .
V ( Z ) = λ 2 π n c Z sin [ 2 π n c ( cos θ min cos θ max ) Z λ ] exp [ 2 π j n c ( cos θ min + cos θ max ) Z λ ]
| V ( Z ) | = λ 2 π n c [ Z ] | sin [ 2 π n c ( 1 cos θ max ) Z λ ] |
R = e j ϕ = { e 0 = 1 for ν < ν S P e j Δ ϕ S P for ν > ν S P .
V ( Z ) = λ 2 n c ν min ν S P e 2 π j ν Z d ν + λ e j Δ ϕ S P 2 n c ν S P ν max e 2 π j ν Z d ν .
V ( Z ) = λ 2 π n c Z sin [ 2 π n c Z λ ( cos θ min cos θ S P ) ] exp [ 2 π j n c Z λ ( cos θ min + cos θ S P ) ] + λ 2 π n c Z e j Δ ϕ S P sin [ 2 π n c Z λ ( cos θ S P cos θ max ) ] exp [ 2 π j n c Z λ ( cos θ S P + cos θ max ) ] .
V ( Z ) = λ 2 π n c Z sin [ 2 π n c Z λ ( 1 cos θ S P ) ] exp [ 2 π j n c Z λ ( 1 + cos θ S P ) ] + λ 2 π n c Z e j Δ ϕ S P sin [ 2 π n c Z λ ( cos θ S P cos θ max ) ] exp [ 2 π j n c Z λ ( cos θ S P + cos θ max ) ] .
ν 1 S P = 1 / Δ Z 1 S P = n c ( 1 cos θ S P ) / λ , ν 2 S P = 1 / Δ Z 2 S P = n c ( 1 + cos θ S P ) / λ , ν 3 S P = 1 / Δ Z 3 S P = n c ( cos θ S P cos θ max ) / λ and ν 4 S P = 1 / Δ Z 4 S P = n c ( cos θ S P + cos θ max ) / λ .
Δ Z S P = λ 2 n c ( 1 cos θ S P ) = Δ Z 1 S P / 2 ,
Δ Z 2 S P = λ n c ( 1 + cos θ S P ) .
V ( Z ) ~ λ 2 π n c Z sin [ 2 π n c Z λ ( 1 cos θ S P ) ] exp [ 2 π j Z ( 1 + cos θ S P ) ] .
| V ( Z ) | ~ λ 2 π n c | Z | | sin [ 2 π n c Z λ ( 1 cos θ S P ) ] | ,
T s ( b , a ) = < s , g b a > = a 1 / 2 s ( t ) g * ( t b a ) d t .
g ( t ) = ( π t 0 2 ) 1 / 4 exp ( 1 2 ( t / t 0 ) 2 + 2 i π ν 0 t ) .
g ( t ) = [ 2 ν 0 π 5 ] 1 / 2 exp [ 2 π 2 ν 0 2 t 2 25 ] exp ( 2 π i ν 0 t ) .

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