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

1. Introduction

Surface plasmon resonance (SPR) has been extensively used since the seventies to probe minute index variations at the interface between a gold film and a dielectric medium [13]. Many geometries including metal films, stripes, nanoparticles, nanorods, holes, slits . . . have been used to sustain surface plasmon polaritons (SPP) which offer a wide range of original properties, such as field enhancement and localization, high surface sensitivity and sub-wavelength localization. This explains why SPP have found applications in a variety of fields such as spectroscopy, nanophotonics, biosensing, plasmonic circuitry, nanolasers, sub-wavelength imaging. SPR is also very popular for the detection of molecular adsorption of small molecules, thin polymer films, DNA or proteins, self-assembled layers . . . [4, 5]. The principle of SPR detection takes advantage of the high sensitivity of SPP to refractive index gradients generated by the adsorption of molecules on a metal substrate. The plasmon wave undergoes a modification of both its amplitude and its phase when it meets an obstacle of different refractive index.

SPP are transverse magnetic (TM or p-polarized) surface waves that propagate along a metal-dielectric interface, typically at visible or infrared wavelengths [3]. SPP are guided waves since both the electric and magnetic amplitudes decay exponentially in the Z direction normal to the interface into both the metal and the dielectric material. Because of dissipative losses in the metal, SPP are also damped in their propagation along the X-direction (parallel to the gold surface and normal to the magnetic field).

More elaborated SPR microscopy techniques with enhanced sensitivity have been proposed, based on Mach-Zehnder interferometry [6,7] or dark-field microscopy [8]. The limitation of all methods using the Kretschmann configuration comes from a few micron lateral resolution due to the guided mode propagation of the plasmon inside the thin metal layer [9]. Coupling light with plasmon modes thanks to oil-immersion [1012] or solid-immersion [13] objective lenses allows to circumvent this difficulty by shaping and confining the plasmon wave laterally to the gold surface. The excitation of SPR with high numerical aperture lenses combined with an interferometric device has pushed down SPR microscopy resolution to sub-wavelength dimensions [1419]. Two methods have been proposed so far: a wide-field SPR microscope [12, 16] and a scanning surface plasmon microscope (SSPM) [14, 15]. We have more specifically focused on the second method during the past ten years [15, 1726].

Conventional SPR machines typically operate in the visible and near-visible wavelength range, from 500 to 800 nm. Surface-plasmon waves in the visible range, characterized by a shallow penetration of less than a few hundreds of nm, are useful for studying monomolecular layers in contact with a sensor surface. This is why SPR is popular for quantitative studies of dynamic interactions in thin biolayers, including molecular recognition of binding events. However, visible-range SPR is not optimal for studying living cells because the cell size may considerably exceed the penetration depth of visible-range surface-plasmon waves. As an alternative, infrared (IR) surface plasmons penetrate much deeper and are more appropriate for studying cells. In particular, the penetration depth of near to mid-IR surface-plasmon waves is a few micrometers, that is just about typical adhering cell height. Traditional glass lenses and optical components can still be used in the very near IR (up to two microns) [27] whereas for larger wavelengths other optical materials such as silicon [28, 29], chalcogenide glass [3032] and ZnS [33, 34] are requested.

In this study, for the first time to our knowledge in the literature, we develop a near IR SSPM (1550 nm), combined with a visible SSPM (632.8 nm). To keep the possibility of high resolution imaging with SPP excitation, at reasonable cost, using the same objective lens, we select 1550 nm as the IR wavelength. We describe the V (Z) response of this microscope in terms of a Fourier transform of the reflected electric field filtered by the objective lens pupil window and we show that the width of this window is a key parameter for maintaining a good microscope contrast since we combine two wavelengths. Because the precise characteristics of the objective lens pupil window are quite impossible to know a priori, it is also important to develop a method to quantify the impact of this filtering on the microscope response. Therefore we propose a wavelet-based space-frequency analysis to decipher the characteristic frequencies of the V (Z) curves and explain how and how much the optical elements can impact the performance of the SPR detection by this microscope

The paper is organized as follows. Section 2 is devoted to the description of the two wavelength SSPM set-up and of the method of blood cell sample preparation. In Section 3 are introduced the principles of SPR and SSPM with their typical Rp(θ) and V (Z) functions. SPR properties in both visible and near IR ranges are described in order to enlighten the capabilities of IR SSPM. The general shape of the V (Z) response is interpreted in terms of a superimposition of different frequencies produced by either the surface plasmon resonance or the confinement of the incident light angles. Since in experimental situations the decoding of these different frequencies is not intuitive, especially in near IR, we use in Section 4 the Morlet wavelet transform to build a space-frequency representation of the V (Z) response. We show that this space-frequency decomposition is well suited to reveal the characteristic frequencies of the experimental V (Z) curves, and this even though the objective lens pupil profile is not known. Finally we apply this wavelet decomposition to theoretical and experimental V (Z) curves obtained from respectively numerical simulations and direct recording from our two-wavelength SSPM. We illustrate the performance of this microscope by comparing a series of images of a blood cell obtained with both visible and near IR lights.

2. Materials and methods

2.1. Optical set-up

Fig. 1 shows the optical set-up combining two SSPMs working at 632.8 nm and 1550 nm respectively, with an inverted microscope stand. On each line we have introduced a heterodyne interferometer equipped with reference mirrors (M1 and M2), beam splitters (BS1 and BS2) and two acousto-optic modulators (AOs). The two polarized laser sources (He-Ne (λ = 632.8 nm) and near IR laser with external cavity (λ = 1550 nm)) are collimated, expanded so that the beam diameters are twice the objective lens (OL) aperture before focusing on the sample. The SSPM image is formed by scanning the sample with a XY piezoelectric stage at a fixed defocus Z. We use a 1.45 NA 60x Olympus objective for both wavelengths. The principle of SPP excitation is analog to Kretschmann configuration [3]: the coupling medium of refractive index n0 is made by an OL, an immersion oil and a coverslip. The gold film sample was prepared by thermal evaporating under vacuum a 35 nm gold layer on a 3 nm chromium layer - (ACM - Application couches minces - Villiers St Frédéric, France). The final dielectric medium with refractive index n3 is air or water (here we use only air). The incoming rays that fall on the glass-gold interface with an angle θSP are coupled to SPP; they propagate at the gold/dielectric interface and reradiate in symmetrical rays back to the objective lens and to the interferometer. The rays which are not coupled to SPP are reflected back into OL without SPP phase retardation. The refractive index n2 and the size of the objects localized at the metal/final media interface lead to local variations of the coupling angle θSP and of the SPP wave vector kSP.

 

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.

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2.2. Cellular sample preparation

Glass coverslips were cleaned in a caustic detergent for one hour at 50°C, and rinsed twice with distilled water (at least 2 hours each) at room temperature; ultimately they were quickly rinsed with ultrapure water (18MΩ.cm−1), absolute ethanol and dried under a nitrogen flow. They were stored in clean compartments and sent to a French company (ACM Application couches -ACM - Villiers St Frédéric) for thermal evaporation of a 3 nm adhesion layer of chromium and 35 nm of gold. If not used immediately, gold coverslips were stored into a vacuum chamber to prevent pollution due to atmospheric impurities. The gold samples where treated by O3 and UV light before being brought to the hematology laboratory for the preparation of the blood extract by blood smearing. EDTA-anticoagulated fresh peripheral blood or bone marrow samples were processed using the whole-blood lysis technique and a drop of the cell suspension was placed at one end of the clean gold-coated coverslip, and the edge of a second slide was drawn across the drop at an angle so that capillary action spreads the drop along the edge. The second slide was then pushed in one smooth motion to the opposite end of the first slide, spreading the drop across the slide to make the smear. This method, although apparently simple, requires some dexterity and several hours of training to produce samples with a single layer of cells and to avoid crashing the cells onto the coverslip.

3. Modeling the scanning surface plasmon microscope

3.1. Reflectivity curves: from visible to IR

Propagation of the plasmon excitation. The wave vector (kSP) for surface plasmon in a Kretschmann configuration can be written as:

kSP=2πλεDεmεD+εm+Δk,
where εD is the dielectric permittivity of the dielectric medium in contact with gold and εm the dielectric permittivity of gold. The first term in Eq. (1) corresponds to the propagation constant of the surface plasmon metal-dielectric guided wave, and Δk accounts for the prism coupling and the thickness of the gold layer. For the plasmon resonance to occur, the propagation constant kEW of the evanescent wave and that of the surface plasmon kSP must be equal along the propagation direction (X):
kEW=2πλncsinθ=kSP=kSP+ikSP,
where nc is the refractive index of the coupling medium (glass in our case).

The fact that kSP is parallel to the gold interface imposes that the incident electric field is p-polarized (transverse mode (TM) for the magnetic field). For each wavelength, the matching condition for SPR is fulfilled for a single angle of incidence θSP, the coupling angle, which increases when decreasing the wavelength λ. The lateral propagation length LX of the plasmon wave is directly related to the inverse of the imaginary part of kSP. If one neglects, in a first approximation, the correction to kSP due to the prism coupling (Δk), the lateral propagation length behaves as:

LX=12kSP=λ2π{εm+εDεmεD}3/2εm2εm.
We have plotted in Fig. 2(a) the evolution of LX with λ in the case of a gold metal film in contact with air. This lateral propagation length increases sharply beyond 600 nm to reach more than 300 μm at 1500 nm. The divergence of this propagation length with λ conveys the idea that the radiative losses of the plasmon wave are decreasing when λ increases; this will be confirmed by a narrowing of the plasmon resonance curve |Rp(θ)| in Fig. 3. Another important conclusion is that the lateral resolution that can be achieved in near-IR wavelength with a prism-coupled plasmon device is worse than with visible wavelengths. This urges those who would like to perform surface plasmon microscopy in IR to use an objective-coupled device to recover the diffraction limit.

 

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].

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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.

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SPR and evanescent waves. Evanescent waves occurring for angles of incidence greater than the critical (total reflection) angle θATR are described by a penetration depth which quantifies the confinement of the field in the vicinity of the interface. This penetration depth depends on the indices of the two media bounding the interface, for instance the index of glass nc and the index of the dielectric medium nD[36]. When the angle of the incident beam θ increases fromθTR to π/2, lZ decreases from ∞ to lZ=λ2πnc2nD2. When SPR is excited, namely when θ = θSP, the penetration depth in the dielectric medium reads:

lZSP=λ2π{|εm+εD|εD2}1/2,
where Z is the direction normal to the gold interface. The evolution of lZSP with λ is shown in Fig. 2(b). We note again that this characteristic attenuation length of the plasmon wave increases nonlinearly with λ, starting from half the wavelength λ at 633 nm (∼ 300 nm) to reach about twice the wavelength λ at 1550 nm (∼ 3 μm). This evolution confirms the strong interest of near-IR plasmon microscopy for the characterization of thicker samples, such as living cells for instance. This attenuation length was computed assuming a prism coupled device. Let us point out that when switching to an objective lens-coupled system, as commonly used in microscopy, this attenuation length can be modulated (increased) with a correct choice of the range of incident angles.

3.1.1. SPR reflectivity curves

Two reflectivity curves of |Rp| versus the incident angle θ are shown in Fig. 3(a) for λ = 632.8 nm (in red) and λ = 1550 nm (in black) respectively. They were computed for a 35 nm layer gold film in contact with air, with a 3 nm adhesion chromium layer, to closely match the experimental conditions that will be described later (see reference [36] for the formula of the transmission and reflection coefficients of a multi-layer system). We selected a 35 nm thickness of gold rather than 45 nm, to increase the signal response of SSPM in near IR range. We observe in Fig. 3 that both reflectivity curves show a drop in between θ = 40 ° and θ = 50 ° (44.26° for λ = 632.8 nm and 42.05° for λ = 1550 nm), corresponding to the plasmon resonance and that the shape of this resonance is definitely much sharper with a near IR excitation, as previously mentioned. We notice also that, below the total internal reflection angle (θ < θATR), the reflectivity in p polarization is higher with near IR than with red wavelength. The plasmon resonance corresponds to a fast variation of the phase of the complex reflectivity Rp, which also depends on the sharpness of the resonance; the sharper the resonance, the closer this phase drop to 2π. We illustrate this property of the surface plasmon resonance on a 3D plot of the real and imaginary components of Rp and the angle θ in Fig. 3(b). At resonance (θ ≳ 40°), the red (visible) and black (near IR) Rp curves make a loop of about one turn, corresponding to this 2π phase drop. We remark also that the red loop is not complete, meaning that the phase drop does not reach 2π, whereas the black loop is quite complete, meaning that with near IR light the phase drop reaches a value closer to 2π.

The angle of plasmon resonance θSP changes with the wavelength λ as:

θSP=arcsin{kSPkc},
where kc = 2πnc/λ. As shown in Fig. 4(a), θSP decreases with λ to get closer and closer to the total reflection angle θATR for near IR wavelengths.

 

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.

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We have pointed out just above that the decrease of the radiative losses of the plasmon wave when increasing λ should materialize in a narrowing of the plasmon resonance curve |Rp(θ)|. Actually the angular width of the surface plasmon resonance is determined by the imaginary part of the surface plasmon wave vector [27]:

ΔθSP=kSPkc2kSP2.
As illustrated in Fig. 4(b), ΔθSP decreases when increasing the wavelength λ above 500 nm.

3.1.2. Sensitivity of the plasmon reflectivity curves with an adsorbed layer

Indeed, the plasmon resonance is very sensitive to adsorbed layers on gold, which explains that it has been extensively used during the past decades for sensing monolayers. It is therefore interesting to compare this sensitivity, in particular the drift of θSP in the presence of an adsorbed layer. To this purpose, we added on the gold film a dielectric layer with optical index 1.4 and thickness e varying from 0 to 250 nm and we computed the modulus of Rp. We reconstructed in Fig. 5, a color coded image of |Rp| (0 corresponding to dark blue, and 1 to red), in the two dimensional space [θ, e].

 

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.

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The drift of the plasmon resonance to larger θ values when the thickness of the added dielectric layer is increased is more important for visible (632.8 nm, Fig. 5(a)) than for near IR (1550 nm, Fig. 5(b)) lights. For both wavelengths, we note that the drop of |Rp| at resonance widens when increasing the dielectric layer thickness. Given the fact that the range of incident angles is limited by the objective numerical aperture, we conclude from the numerical results reported in Fig. 5 that the smallest thickness variations will be better detected from the significant drift in θSP observed with the visible light. However, beyond 100 nm, the resonance drop fades and it crosses the limit angle of the objective lens (arcsin(NA/nc) ∼ 73° with NA=1.45 and nc = 1.5151), making impossible the use of SPR excitation mode for microscopic imaging. Meawhile, the slower drift of the drop of |Rp| with the dielectric thickness e in near IR excitation allows us to characterize much thicker samples, typically up to 700 nm thickness. In the case of near IR excitation light, for dielectric thickness larger than 700 nm, the plasmon resonance angle θSP of a 35 nm gold film in air saturates at about 70°, inside the range of angle afforded by a 1.45 NA objective lens.

It is also very instructive to investigate how the phase drop at plasmon resonance varies in the presence of an adsorbed dielectric layer. We use in Fig. 6 the same 3D representation of the complex reflectivity Rp, with the incident angle θ along the vertical axis as in Fig. 3(b). This representation shows that in visible light, the phase drop at plasmon resonance starts from a value close to 2π for e = 0 nm and progressively decreases to values close to π for e = 250 nm. For near IR light, the phase drop changes much slower than before when increasing e, showing that the plasmon resonance sensor keeps its high sensitivity for a wider range of thickness values. Thus, depending on the thickness of the sample to be characterized, the wavelength of operation will have to be optimized, the thicker the sample, the larger the wavelength.

 

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.

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3.2. V (Z) curves

SSPM is a combination of an heterodyne interferometer, a high numerical aperture objective lens and a (X, Y, Z) piezoelectric scanning device (Fig. 1). It is commonly used to record the so-called V (Z) response [14,15]; this is illustrated in Fig. 7 where are shown two typical experimental |V (Z)| curves obtained with our two-wavelength optical device. We notice immediately that the interpretation of these curves for positive Z values, is not at all intuitive since they look like a mixture of non stationary oscillatory behavior that cannot be separated by simple Fourier analysis. We will therefore need an adapted space-frequency analysis to identify the underlying frequencies and to interpret them in terms of plasmon mode frequencies. V (Z) is an integral over the radial θ (θmin, θmax) and azimuthal φ (0, 2π) angles of the backward reflected electric field after plasmon excitation at the gold dielectric interface. The reflected field is described by Rp (resp Rs) in p (resp. s) polarization configuration. This method is directly inspired from scanning acoustic microscopy [14, 37, 38].

V(Z)=θminθmaxP2(θ)𝒬(θ)e2j(2πnc/λ)Zcosθsinθdθ,
where P is the pupil response of the objective lens, nc is the index of the objective lens and the matching index oil. 𝒬(θ) is proportional to the back reflected field, integrated over the radial angle φ. This quantity depends on the light polarisation; for p polarized light it reads:
𝒬(θ)=02πRp(sinθ)cos2φdφ=πRp(sinθ);
for s polarized light, it takes the following alternative form:
𝒬(θ)=02πRs(sinθ)sin2φdφ=πRs(sinθ).

Ideally, this pupil function P should be as flat as possible (constant). In our experiment it will be modeled by a rectangular window with smooth gaussian borders of the form (Fig. 8):

P(cosθ)=[1exp(cosθ22σ2)].
As far as the analytical modeling is concerned, we will assume (for the sake of simplicity) that P = 1.

 

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).

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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.

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With the following change of variables [39]:

ν=(2nc/λ)cosθ,dν=(2nc/λ)sinθdθ,
the V (Z) function takes the form of the Fourier transform of the reflected field:
V(Z)=λ2ncνminνmaxP2(ν)𝒬(ν)e2πjνZdν,
where
νmin=(2nc/λ)cosθmaxandνmax=(2nc/λ)cosθmin.
For an objective lens with numerical aperture NA, θmax = arcsin(NA/nc) and νmin=(2π/λ)nc2NA2. Without diaphragming the light, we have θmin = 0, and νmax = 2nc/λ. We define a new pupil function such as = 0 outside the angular interval [θmin = 0, θmax], and = P inside this interval. Eq. (12) can then be reexpressed as:
V(Z)=λ2ncP¯2(ν)𝒬(ν)e2πjνZdν.
Now if we assume that the reflection of light is constant on the angular domain of the optical system (𝒬(θ) = 1), then V (Z) corresponds to the Fourier transform of 2 :
V(Z)=λ2ncP¯2(ν)e2πjνZdν,
which turns out to reduce to the following integral on the [νmin, νmax] domain:
V(Z)=λ2nc2ncλcosθmax2ncλcosθmine2πjνZdν.
This integral can be computed analytically:
V(Z)=λ2πncZsin[2πnc(cosθmincosθmax)Zλ]exp[2πjnc(cosθmin+cosθmax)Zλ]
Two spatial frequencies show up in V (Z): ν1 = 1/ΔZ1 = nc(cosθmin − cosθmax)/λ and ν2 = 1/ΔZ2 = nc(cosθmin + cosθmax)/λ. Taking the numerical aperture of the objective lens that will be used in the experiments: NA= 1.45, and θmin = 0, we get the set of values for the two spatial frequencies ν1 and ν2 reported in Table 1.

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

Note that the modulus of V (Z)

|V(Z)|=λ2πnc[Z]|sin[2πnc(1cosθmax)Zλ]|
is the absolute value of a normalized sinc function (sin(t)/t). The period of |V (Z)| is ΔZ1/2 ≃ 299.62 nm with λ = 632.8 nm (2ν1 ∼ 3.4 μm−1) and ≃ 694.5 nm with λ = 1550 nm (2ν1 ∼ 1.44 μm−1).

As shown in Fig. 9, when the pupil borders are smoothed (Fig. 8), the periodic modulation with frequency ν1 of |V (Z)| vanishes to leave a simple 1/Z decay. One would then think that the smoothest P would be best for surface plasmon microscopy because it would avoid the mixing of the pupil window frequencies with those of the plasmon resonance itself. In practice, there is a compromise to reach between this smoothing and the fact that there should be enough light power in the range of angles corresponding to θSP. A similar issue has been recently raised by Zhang et al[40] who proposed to introduce a spatial light modulator confocal pinhole in a wide field SPR microscope to control the microscope pupil function. Although this system could be a more stable and compact design than the heterodyne interferometer reported here, its loss of signal to noise ratio could be detrimental to its performance in near IR.

 

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.

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3.3. V (Z) response to a phase drop

The phase of the reflectivity undergoes a very sharp drop at plasmon resonance. Its impact on V (Z) can be computed assuming that the reflectivity modulus |Rp| = 1 everywhere, except when θ = θSP, where |Rp(θSP)| tends to zero at resonance (Fig. 3(a)). This is fairly realistic in case of no radiative metal loss. The phase drop can be described by a Heaviside function, which means that below resonance ϕ = 0 and above resonance ΔϕSP = ±2π +εϕ. This small quantity εϕ is important since it describes the sharpness of the plasmon resonance. Accordingly:

R=ejϕ={e0=1forν<νSPejΔϕSPforν>νSP.

Then the integral in the definition (Eq. (12)) of V (Z) splits in two integrals:

V(Z)=λ2ncνminνSPe2πjνZdν+λejΔϕSP2ncνSPνmaxe2πjνZdν.
We get the sum of two modulated sinc functions, with a phase term in factor of the second one:
V(Z)=λ2πncZsin[2πncZλ(cosθmincosθSP)]exp[2πjncZλ(cosθmin+cosθSP)]+λ2πncZejΔϕSPsin[2πncZλ(cosθSPcosθmax)]exp[2πjncZλ(cosθSP+cosθmax)].

Taking cosθmin = 1, we obtain:

V(Z)=λ2πncZsin[2πncZλ(1cosθSP)]exp[2πjncZλ(1+cosθSP)]+λ2πncZejΔϕSPsin[2πncZλ(cosθSPcosθmax)]exp[2πjncZλ(cosθSP+cosθmax)].
Now V (Z) involves four spatial frequencies:
ν1SP=1/ΔZ1SP=nc(1cosθSP)/λ,ν2SP=1/ΔZ2SP=nc(1+cosθSP)/λ,ν3SP=1/ΔZ3SP=nc(cosθSPcosθmax)/λandν4SP=1/ΔZ4SP=nc(cosθSP+cosθmax)/λ.
Given the plasmon resonance angle θSP at 632 nm and 1550 nm, the values of these four spatial frequencies for visible and near IR lights are reported in Table 2.

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.

Let us point out the presence of a sinc function in the first term in Eq. (22), so that the corresponding period of its modulus is the half period of the sine:

ΔZSP=λ2nc(1cosθSP)=ΔZ1SP/2,
which corresponds to the formula originally proposed by Somekh et al in 2000 [14]. But we have also a second spatial period that occurs in the real and imaginary parts of V (Z):
ΔZ2SP=λnc(1+cosθSP).
If we take θSP = 44.26°, nc = 1.5151, for visible light, then we obtain the spatial periods ΔZ1SP/2 ∼ 735.8 nm (corresponding to 2ν1SP ∼ 1.35 μm−1) and ΔZ2SP ∼ 243.4 nm (corresponding to ν2SP ∼ 4.11 μm−1). The period of |V (Z)| corresponding to the resonance (735.8 nm) is thus much larger than the period of the sinc function defined by the objective aperture in Table 1 (294 nm = 1/2ν1). What is important to remind is the fact that when θSP increases the period ΔZSP decreases. If the objective lens pupil window is smoothed to damp the sinc modulations due to the θmax boundary, then only the first term in Eq. (22) contributes significantly to the V (Z) function:
V(Z)~λ2πncZsin[2πncZλ(1cosθSP)]exp[2πjZ(1+cosθSP)].
Then, from the V (Z) modulus:
|V(Z)|~λ2πnc|Z||sin[2πncZλ(1cosθSP)]|,
it will be easy to discriminate and extract the spatial frequency 2ν1SP related to surface plasmon resonance.

4. Deciphering the frequencies of V (Z) curves using a Morlet wavelet transform

We have shown in Section 3 that the V (Z) response curve involves a superimposition of different frequencies produced by either the surface plasmon resonance or the confinement of the incident angles. In experimental situations it is very difficult when the light source wavelength is varied to know precisely the evolution of these frequencies and to separate surface plasmon modes from other effects coming from the optical device itself. In this section, we apply a wavelet-based space-frequency method to achieve this decomposition. The continuous wavelet transform(CWT) is a mathematical technique introduced in signal analysis in the early 1980s [41,42]. Since then, it has been the subject of considerable theoretical developments and practical applications in a wide variety of fields [4358]. Some optical wavelet device has been designed that performs the CWT thanks to Fourier windowing principles [59, 60].

The CWT amounts to project an arbitrary signal s(t) onto a family of functions {gba(t)}, where b ∈ ℜ is the time parameter and a ∈ ℜ+ the scale parameter [44, 52, 53]:

Ts(b,a)=<s,gba>=a1/2s(t)g*(tba)dt.
The complex analyzing wavelet introduced by Morlet [61, 62] was inspired from Gabor analysis. It has the form of a modulated Gaussian function:
g(t)=(πt02)1/4exp(12(t/t0)2+2iπν0t).
The Morlet wavelet satisfies the admissibility condition: ∫ gba(t)dt = 0. As commonly used, this condition can be enforced to a very good approximation when 2πt0ν0 = 5.4285. In this study we fix 2πν0t0 = 5, so that the Morlet wavelet takes the following form:
g(t)=[2ν0π5]1/2exp[2π2ν02t225]exp(2πiν0t).

4.1. Morlet space-frequency maps of V (Z) curves for visible light

The Morlet decomposition of the V (Z) function is performed in Fourier space, by a direct product of the Fourier transform of the analyzing Morlet wavelet with the Fourier transform of ln|V (Z)|. Taking the reverse Fourier transform gives directly the wavelet transform at given scale a. This operation is repeated for different values of the frequency ν = 1/a (in μm−1), from which we construct a space-frequency map using a color coding from blue (minima value of the wavelet transform modulus) to red (maxima value of the wavelet transform modulus). Such a space-frequency map is shown in Fig. 10(a) for the V (Z) function computed for a gold film of 35 nm, with a sublayer of 3 nm of chromium, and a constant pupil window P = 1 for 0 < θ < θmax = arcsin(1.45/1.5151) ∼ 73° in visible light (λ = 632.8 nm). We recognize in this representation the frequency 2ν1 ∼ 3.4 μm−1 corresponding to the sharp pupil borders at θmax; it manifests as a wide red band for negative Z values and for Z larger than 15 μm. In the remaining range of Z values emerge two other spatial frequencies, one in between 1 and 2 μm−1 and the other one around 2 μm−1. The smaller one corresponds to 1/ΔZSP = 2ν1SP ∼ 1.36 μm−1 whereas the larger one corresponds to 2ν3SP ∼ 2.04 μm−1 (Tables 1 and 2) and results from some mixing of the plasmon frequency with the pupil border frequency (Eq. (23)). When the objective pupil is smoothed (Figs 10(b) and 10(c)), the influence of the pupil borders progressively vanishes and we recover the spatial frequency given by the plasmon resonance only. This space-frequency representation shows also that it is necessary to have a strong smoothing of the pupil function of the objective lens to recover the characteristic spatial frequency of the plasmon resonance for Z values smaller than 5 μm.

 

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.

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4.2. Morlet space-frequency maps of V (Z) curves for near IR light

In near IR light, the same scenario is observed in the space-frequency maps when the pupil window becomes smoother. In Fig. 11(a), we see mainly the frequency 2ν1 ∼ 1.44 μm−1 (red horizontal band) corresponding the sharp boundary of the pupil (Table 1). This band is very flat for negative Z values, and shows extra modulations for positive Z values that originate from the presence of the plasmon resonance. The band corresponding to the plasmon frequency νSP = 2ν1SP ∼ 0.498 μm−1 is quite imperceptible if the pupil window is not smoothed enough (Fig. 11(a) and 11(b)). Smoothing further the pupil window cancels completely the 1.44 μm−1 modulation and strengthens the band corresponding to νSP that definitely emerges from the background (Fig. 11(c)).

 

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.

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4.3. Experimental data

In Fig. 12 are shown the experimental V (Z) curves obtained with our two-wavelength SSPM device for a 35 nm gold film prepared by thermal evaporation on a 3 nm chromium layer. The two ln|V (Z)| signals are plotted at the bottom of Figs 12 (a) and (b). We notice that the |V (Z)| obtained for visible wavelength λ = 632.8 nm reveals the characteristic spatial frequency of plasmon resonance for Z values lower than 5 μm. The strong similarity observed between the Morlet space-frequency maps reconstructed from the experimental (Fig. 12(a)) and the numerical (Fig. 10(c)) V (Z) functions suggests a strong experimental smoothing of our objective lens pupil.

 

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

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The experimental V (Z) curve recorded with near IR light (λ = 1550 nm), for the same gold film sample, the same day, with the same objective lens, the same matching index liquid and without disassemblying the optical set-up (since both visible and near IR light can be run simultaneously) is shown in Fig. 12(b). It turns out to be very different from the one obtained with visible light, and this not only for the values of its spatial frequencies, but also for their extensions in Z. In particular, two spatial frequencies are evidenced by the Morlet space-frequency map, a lower one around 0.4 μm−1, likely corresponding to the plasmon resonance νSP, and a larger one, close to 1 μm−1 that probably corresponds to the objective pupil itself (2ν1). When comparing the experimental (Fig. 12(b)) and the numerical (Fig. 11) Morlet space-frequency maps, we can conclude that in near IR light, the objective lens has a numerical aperture that is smaller than the value given for visible light, NAIR = 1.5 * sin(arccos(1 − 1.550/(2 * 1.5))) ∼ 1.31 < 1.45, and its pupil function is flatter than in visible light. This apparent decrease of the objective lens numerical aperture in the near IR probably results from the fact that this Olympus M Plan Apochromat lens was aberration corrected for blue, green and red wavelengths and not for near IR wavelengths.

4.4. Application to blood cell imaging

The two sets of four images shown in Fig. 13 were captured with the two-wavelength SSPM set-up from blood cell smearing on the gold film (Section 2). The imaged cell was selected because, on the one hand it showed adhesion filopodia that strongly adhered on the gold surface (Figs 13(a)–13(c)), and on the other hand it had also a small filamentous body that came out of the gold plane surface (Figs 13(a’)–13(c’)). Since the SSPM offers the possibility to scan the sample at different penetration depths thanks to the defocus parameter Z, four values of Z are represented in Fig. 13 that correspond to focusing inside the glass (Z < 0, Figs 13(a), 13(a’)), on the gold film (Z ∼ 0, Figs 13(b), 13(b’)), on the cell media (Z > 0, Figs 13(c), 13(c’)) and on air above the cell surface (Z > 0, Figs 13(d), 13(d’)).

 

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.

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The comparison of the visible and near IR images confirms that the integration depths for the two wavelengths are very different. For instance, the picture obtained for λ = 632.8 nm and Z = 1.25 μm (Fig. 13(c)) must be compared to the picture obtained with near IR λ = 1500 nm and Z = −0.5 μm (Fig. 13(b’)). The depth of integration of the cell body through the evanescent field afforded with the near IR light is actually much larger; in particular it allows the observation of a filamentous body on the top of the cell (Fig. 13(d’)), that cannot be captured with the visible light (Fig. 13(d)). However, for capturing cell bodies that are firmly anchored on the gold surface, the visible light SSPM (Fig. 13(b)) is more appropriate than the near IR SSPM (Fig. 13(b’)). For larger Z defocus values, the near IR SSPM can still provide images that are not limited by diffraction effects. The near IR image in Fig. 13(d’) shows some granulations on the cell border that could not be observed with visible light in Fig. 13(d) since they are protruding outside the range of the visible light evanescent field. The combination of both these wavelengths gives therefore instructive and complementary views of the cell, without the need of any external dying agent that could disturb and spoil its structure.

5. Conclusion and future prospects

In this paper, we have shown that high resolution scanning surface plasmon microscopy can be carried out in near IR range, with the same objective lens as in visible range, providing a deeper scanning in the Z direction. This microscope is more suitable to investigate thicker samples, such as cells and tissues. We have shown that the two variables ν and Z defined in Eq. (12) are conjugated entities in this microscope. As a consequence, when the surface plasmon resonance is sharper in ν = ((2nc/λ)cosθ), as observed in near IR light, the V (Z) function is less localized in Z, which allows the characterization of thicker samples.

We have illustrated the performance of this dual wavelength microscope to image a lymphocyte cell. The implementation of a space-frequency analysis of the response function V (Z) of this microscope has proved effective in deciphering the underlying spatial frequencies that can be related to characteristic angles such as the surface plasmon resonance angular position and the angular boundaries defined by the numerical aperture of the objective lens. The advantage of this wavelet decomposition is its ability to perform a local analysis of the frequency response, which cannot be done with classical Fourier method. This space-frequency analysis has enlightened the crucial role played by the shaping the light profile (in particular the pupil of the objective lens) in surface plasmon microscopy. In particular, as regards to previous studies where annular diaphragms were used to shape the incoming light [10, 63, 64], it could be interesting to select the width and the shape of the diaphragm with the help of this wavelet based space-frequency decomposition to optimize the contrast of the surface plasmon response depending on the index and thickness of the sample.

Finally, the operation of this dual wavelength microscope in liquid media is an experimental challenge, since in a liquid medium the plasmon resonance angles increase and get closer to the angular limits of the objective lens. The superimposition of the corresponding spatial frequencies makes then harder to separate them. We are currently testing two methods to achieve this goal, namely the introduction of an objective lens with a larger numerical aperture (which is more expensive) and the modification of the index of the glass/gold/dielectric interface to recover resonance curves within an affordable angular range compatible with lower numerical aperture objective lenses.

Acknowledgments

We are very endebted to Lyon Science Transfert (Lyon University) for financial support (project L659). We acknowledge the funding of Région Rhône Alpes under a CIBLE project (CIBLE2011). This work has been supported by the French Agency for Research (ANR) under contract ANR- AA-PPPP-005, EMMA 2011 program. We are also endebted to the Interdisciplinary Programs of CNRS (soutien prise de risque) which funded this work in 2009.

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References

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  1. A. Otto, “Excitation of nonradiative surface plasmon waves in silver by the method of frustrated total reflection,” Z. Angew. Phys. 410, 398–410 (1968).
  2. E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excitated by light,” Z. Natur-forsch., A: Phys. Sci. 23, 2135 (1968).
  3. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, Heidelberg, 1988).
  4. 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]
  5. J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377(3), 528–539 (2003).
    [CrossRef] [PubMed]
  6. 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, A 85(1–3), 189–193 (2000).
  7. A. G. Notcovich, V. Zhuk, and S. G. Lipson, “Surface plasmon resonance phase imaging,” Appl. Phys. Lett. 76(13), 1665–1667 (2000).
    [CrossRef]
  8. A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174(January), 151–155 (2000).
    [CrossRef]
  9. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B: Condens. Matter 33(8), 5186–5201 (1986).
    [CrossRef]
  10. H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. A 15(4), 1381–1386 (1998).
    [CrossRef]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter 81(23),  235,419 (2010).
    [CrossRef]
  18. J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A 27(3), 450–457 (2010).
    [CrossRef]
  19. 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. Express 19(7), 2829–2836 (2011).
    [CrossRef]
  20. J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt. 7(9), 472–478 (2005).
    [CrossRef]
  21. 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]
  22. 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 China 4(4), 469–480 (2009).
    [CrossRef]
  23. 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]
  24. 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]
  25. 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]
  26. 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]
  27. 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]
  28. 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]
  29. 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. Express 17(4), 2255–2263 (2009).
    [CrossRef] [PubMed]
  30. 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, B 130(2), 771–776 (2008).
    [CrossRef]
  31. 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]
  32. 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]
  33. 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]
  34. 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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  37. A. Atalar, “An angular-spectrum approach to contrast in reflection acoustic microscopy,” J. Appl. Phys. 40, 5130–5139 (1978).
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  38. A. Atalar, “A physical model for acoustic signatures,” J. Appl. Phys. 50, 8237–8239 (1979).
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  39. C. Ilett, M. Somekh, and G. Briggs, “Acoustic microscopy of elastic discontinuities,” Proc. Roy. Soc. Lond. Ser. A 393, 171–183 (1984).
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  40. B. Zhang, S. Pechprasarn, J. Zhang, and M. G. Somekh, “Confocal surface plasmon microscopy with pupil function engineering.” Opt. Express 20(7), 7388–97 (2012).
    [CrossRef] [PubMed]
  41. P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration 23, 85–102 (1984).
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  42. 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).
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  43. J. M. Combes, A. Grossmann, and P. Tchamitchian, eds., Wavelets (Springer, Germany, 1989).
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  44. I. Daubechies, Ten Lecture on Wavelets (SIAM, 1992).
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  45. Y. Meyer, ed., Wavelets and their Applications (Springer, Germany, 1992).
  46. M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, eds., Wavelets and their Applications (Jones and Bartlett, Boston, 1992).
  47. P. Flandrin, Temps-Fréquence (Hermès, France1993).
  48. Y. Meyer and S. Roques, eds., Progress in Wavelets Analysis and Applications (Editions Frontières, Gif-sur-Yvette, 1993).
  49. 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).
  50. A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1–2), 232–275 (1995).
    [CrossRef]
  51. G. Erlebacher, M. Y. Hussaini, and L. M. Jameson, eds., Wavelets: Theory and Applications (Oxford University, 1996).
  52. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, USA, 1998).
  53. B. Torresani, Analyse Continue par Ondelettes (Editions de Physique, France, 1998).
  54. 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 Physics 126, 1–92 (2003).
    [CrossRef]
  55. 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]
  56. J. P. Antoine, R. Murenzi, P. Vandergheynst, and S. T. Ali, Two-Dimensional Wavelets and their Relatives (Cambridge University, UK, 2008).
  57. A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia 3, 4103 (2008).
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  58. 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 Reports 498(2–3), 45–188 (2011).
    [CrossRef]
  59. 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]
  60. 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]
  61. J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics 47(2), 203–221 (1982).
    [CrossRef]
  62. J. Morlet, G. Arensz, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part II: Sampling theory and complex waves,” Geophysics 41(2), 222–236 (1982).
    [CrossRef]
  63. 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).
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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. Express 20(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 Reports 498(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. Express 19(7), 2829–2836 (2011).
[CrossRef]

2010 (4)

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]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A 27(3), 450–457 (2010).
[CrossRef]

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter 81(23),  235,419 (2010).
[CrossRef]

2009 (10)

R. Vander and S. G. Lipson, “High-resolution surface-plasmon resonance real-time imaging,” Opt. Lett. 34(1), 37–39 (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. Express 17(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]

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]

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]

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 China 4(4), 469–480 (2009).
[CrossRef]

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]

2008 (2)

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, B 130(2), 771–776 (2008).
[CrossRef]

A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia 3, 4103 (2008).
[CrossRef]

2007 (1)

2005 (2)

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]

J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt. 7(9), 472–478 (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 Physics 126, 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, A 85(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]

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]

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]

1998 (1)

H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. A 15(4), 1381–1386 (1998).
[CrossRef]

1995 (1)

A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(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. Matter 33(8), 5186–5201 (1986).
[CrossRef]

1984 (3)

C. Ilett, M. Somekh, and G. Briggs, “Acoustic microscopy of elastic discontinuities,” Proc. Roy. Soc. Lond. Ser. A 393, 171–183 (1984).
[CrossRef]

P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration 23, 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,” Geophysics 47(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,” Geophysics 41(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. Matter 6(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, B 130(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, B 130(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,” Geophysics 47(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,” Geophysics 41(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. Express 19(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 Reports 498(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. A 27(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. Matter 81(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 China 4(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 Reports 498(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,” Scholarpedia 3, 4103 (2008).
[CrossRef]

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[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 Physics 126, 1–92 (2003).
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A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1–2), 232–275 (1995).
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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).
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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).
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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 Reports 498(2–3), 45–188 (2011).
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A. Arneodo, B. Audit, and P. Kestener, “Wavelet-based multifractal analysis,” Scholarpedia 3, 4103 (2008).
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A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1–2), 232–275 (1995).
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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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A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174(January), 151–155 (2000).
[CrossRef]

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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. Express 19(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).
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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. Matter 81(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. A 27(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 China 4(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]

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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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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).
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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, B 130(2), 771–776 (2008).
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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, B 130(2), 771–776 (2008).
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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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[CrossRef]

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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).
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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 Physics 126, 1–92 (2003).
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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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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, B 130(2), 771–776 (2008).
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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. Express 19(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. Matter 81(23),  235,419 (2010).
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J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A 27(3), 450–457 (2010).
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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 China 4(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).
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J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt. 7(9), 472–478 (2005).
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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).
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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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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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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).
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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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A. N. Grigorenko, A. A. Beloglazov, and P. I. Nikitin, “Dark-field surface plasmon resonance microscopy,” Opt. Commun. 174(January), 151–155 (2000).
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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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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 China 4(4), 469–480 (2009).
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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, B 130(2), 771–776 (2008).
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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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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).
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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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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 Reports 498(2–3), 45–188 (2011).
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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 China 4(4), 469–480 (2009).
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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).
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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 Physics 126, 1–92 (2003).
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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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Appl. Opt. (1)

Appl. Phys. Lett. (3)

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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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).
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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 China 4(4), 469–480 (2009).
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P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration 23, 85–102 (1984).
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J. Morlet, G. Arenss, E. Fourgeau, and D. Giard, “Wave propagation and sampling theory-Part I: Complex signal and scattering in multilayered media,” Geophysics 47(2), 203–221 (1982).
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[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).
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G. Stabler, M. G. Somekh, and C. W. See, “High-resolution wide-field surface plasmon microscopy,” J. Microsc. 214(Pt 3), 328–333 (2004).
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J. Elezgaray and F. Argoul, “Topography reconstruction from surface plasmon resonance data,” J. Opt. A: Pure Appl. Opt. 7(9), 472–478 (2005).
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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. Opt. Soc. Am. A (2)

H. Kano, S. Mizuguchi, and S. Kawata, “Excitation of surface-plasmon polaritons by a focused laser beam,” J. Opt. Soc. Am. A 15(4), 1381–1386 (1998).
[CrossRef]

J. Elezgaray, T. Roland, L. Berguiga, and F. Argoul, “Modeling of the scanning surface plasmon microscope,” J. Opt. Soc. Am. A 27(3), 450–457 (2010).
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H. Kano and W. Knoll, “A scanning microscope employing localized surface-plasmon-polaritons as a sensing probe,” Opt. Commun. 182(August), 11–15 (2000).
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Opt. Express (3)

Opt. Lett. (6)

Phys. Rev. A: At. Mol. Opt. Phys. (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).
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Phys. Rev. B: Condens. Matter (3)

T. Roland, L. Berguiga, J. Elezgaray, and F. Argoul, “Scanning surface plasmon imaging of nanoparticles,” Phys. Rev. B: Condens. Matter 81(23),  235,419 (2010).
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E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64(7), 745–748 (1990).
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A. Arneodo, E. Bacry, and J. F. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1–2), 232–275 (1995).
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Physics Reports (1)

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 Reports 498(2–3), 45–188 (2011).
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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, A 85(1–3), 189–193 (2000).

Sens. Actuators, B (1)

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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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