1. INTRODUCTION

Optical fluorescence microscopy (OFM) is a widely used imaging tool in biology to study tissues at the scale of cells down to single molecules [1, 2, 3]. Unfortunately, the resolution of this far-field imaging technique is fundamentally limited by the diffraction process. Point scanning confocal microscopes obtain at best a lateral resolution (perpendicular to the optical axis) of $\lambda ∕3$, where λ is the illumination wavelength, i.e., several hundreds of nanometers [4]. Hence, improving the spatial resolution of far-field fluorescence microscopes below $100\mathrm{nm}$ has become a major challenge in the last ten years and has spurred the emergence of many innovative techniques [5].

Structured illumination microscopy has shown its potential for ameliorating the resolution in both the lateral (perpendicular to the optical axis) and the axial (along the optical axis) directions while keeping a wide-field configuration [6]. In this approach, the sample is illuminated by a periodic light grid stemming from the interference of two or three coherent beams (standing-wave pattern). Images of the sample are recorded for different positions of the grid, and the fluorescence density is retrieved numerically using an inversion algorithm [7, 8]. The smaller the period of the light grid, the better the resolution. This technique has been extended to total-internal-reflection fluorescence microscopy (TIRFM), in which the excitation field is an evanescent wave obtained through the total internal reflection of the beam at a glass/air interface [9, 10]. This configuration is well adapted to surface imaging inasmuch as only the fluorescent markers close to the interface are strongly illuminated. TIRFM is widely used in biological studies, especially for the observation of cell plasma membranes or biomolecular dynamics [11]. The higher resolution of the standing-wave TIRFM (SW-TIRFM) stems from the fact that the excitation light grid has a period of $(\lambda ∕2n_s)\sin \theta$, where $n_s$ is the refractive index of the high-index material whence the illumination comes and θ is the angle of incidence. Fundamentally, the lateral and axial resolution of SW-TIRFM is monitored by the available high refractive index of the substrate, typically 1.45 for a convenient prismless, objective-launched TIRFM. Recently, this method has been shown to achieve a lateral resolution of approximately $100\mathrm{nm}$ [10] for an illumination wavelength of $500\mathrm{nm}$ while keeping the $100\mathrm{nm}$ axial resolution of conventional TIRFM.

To further improve the resolution, it is possible to take advantage of the saturation process that monitors the fluorescence at high power. The saturated structured illumination microscope [12, 13] has been shown to exhibit a resolution of the order of a few tens of nanometers. Yet, this nonlinear phenomenon requires high-power pulsed lasers with an increased risk of photobleaching and photodisruption in comparison with conventional fluorescence imaging [6, 12]. More recently, it has been proposed to ameliorate the resolution by reducing the period of the structured illumination below that of the standing-wave pattern, i. e., below the diffraction limit. This can be done by replacing the conventional glass slide of the microscope by a periodically nanostructured glass slide, (hereafter called grating) [14]. The grating is designed so as to generate a movable highly contrasted light grid with a period smaller than half the excitation wavelength. The interest of this approach is that it relies only on linear processes and thus can be implemented with low-power lasers and with any fluorescent markers.

In this work, we present an extended numerical study of grating-assisted TIRFM. We compare the resolution of TIRFM using homogeneous illumination, standing-wave illumination, homogeneous illumination on a grating substrate, and standing-wave illumination on a grating substrate. We describe the optimization process of the grating and present an efficient inversion algorithm for retrieving the fluorophore distribution.

2. SIMULATION OF A TIRFM EXPERIMENT

We consider a sample made of one layer of fluorescent beads of diameter $20\mathrm{nm}$ deposited on a substrate, either a standard glass slide or a grating. The sample is placed at the focal object plane of an immersed objective with numerical aperture $\mathit{NA}=1.45$; see Fig. 1 . The intensity that is recorded at the image plane of the microscope can be simulated by

3. DESCRIPTION OF THE ILLUMINATION CONFIGURATIONS

We consider four different illumination configurations, depicted in Fig. 2 . In the first one, Fig. 2a, the sample is deposited on a homogeneous glass slide and is illuminated from below by one plane wave with polar angle ${\theta }_{\mathrm{inc}}=65\mathrm{deg}$ and 12 different azimuthal angles $\varphi =n\pi ∕6$, $n=(0,\cdots ,11)$. The polar angle is chosen beyond the critical angle at the glass/air interface so that the light intensity decays exponentially as one moves away from the glass slide. Whatever the azimuthal angle, one obtains a uniform excitation light on the substrate. We record 12 images to be consistent with the other measurements and, in particular, to have the same sensitivity to noise.

In the second case, the sample is illuminated by two interfering plane waves that create a standing-wave pattern [10, 16] [see Fig. 2b]. The polar incident angles of the two beams are ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$, and we take four different azimuthal angles $\varphi =n\pi ∕4$, $n=(0,1,2,3)$. For each incidence, the relative phase ψ between the two beams is varied three times $\psi =n2\pi ∕3$, $n=(0,1,2)$ so that eventually 12 images are recorded. Basically, the excitation light at $z=0$, just above the substrate, can be modeled by

In the third configuration, the sample is deposited on an optimized grating and is illuminated by one plane wave with electric field parallel to the substrate (TE polarization), polar angle ${\theta }_{\mathrm{inc}}=65\mathrm{deg}$, and 12 different azimuthal angles $\varphi =n\pi ∕6$, $n=(0,\cdots ,11)$ [see Fig. 2c]. The electric field above a grating illuminated by a plane wave is pseudoperiodic with the same periodicity as the grating. The light intensity at $z=0$ just above the grating can be modeled by the square of a Rayleigh series [17],

In the fourth configuration, Fig. 2d, the sample is deposited on the same grating as previously and is illuminated by two interfering plane waves with polar angle ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$, three different azimuthal angles $\varphi =\pi ∕2+n\pi ∕3$, $n=(0,1,2)$, and four different relative phases $\psi =n\pi ∕2$, $n=(0,1,2,3)$. In this case, the intensity just above the grating is no longer periodic. Using the symmetry of properties of the structure, the intensity obtained for ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$ and $\varphi =\pi ∕2$ can be modeled by

4. DESCRIPTION OF THE INVERSION ALGORITHM

In standing-wave microscopy, the inversion algorithm that permits recovery of the density of fluorescence from the various images is usually based on a noniterative procedure, the first step of which consists of the calculation of the two-dimensional Fourier transform ${\tilde{I}}_l^{\mathrm{mes}}$ of the recorded intensities. It takes advantage of the phase variations of the Fourier transform of the illumination ${\tilde{E}}_l$, Eq. (3), obtained through the three translations of the standing-wave pattern, to retrieve, through a one-to-one correspondence, the Fourier amplitude of the density of fluorophores $\tilde{O}$ on an extended Fourier domain [7, 10, 20]. Unfortunately, when the sample is deposited on a grating and illuminated by one plane wave, the different illumination patterns are obtained by changing the incident azimuthal angle, and there is no additional translation of the pattern as in standing-wave microscopy. Hence, it is impossible to use the same algorithm to infer $\tilde{O}$ from ${\tilde{I}}_l^{\mathrm{mes}}$. Thus, we derived a very general iterative inversion scheme to reconstruct the fluorophore density. We checked that this conjugate-gradient algorithm yielded the same result as the Fourier inversion procedure in the standard standing-wave configuration.

We first define a two-dimensional investigation domain Ω at the surface of the glass slide (or of the grating), and we estimate the fluorophore density $\widehat{O}$ at the nodes of a regular meshing of Ω so as to minimize the cost functional,

The problem is now stated as finding the density of fluorophores such that the associated intensity, computed from Eq. (7), matches the measured one, $I_l^{\mathrm{mes}}$. We propose in the present paper an iterative construction of $\widehat{O}$ using the following recursive relationship:

The scalar weight ${\alpha }_n$ is determined at each iteration step n by minimizing the cost functional $\mathcal{F}$ given by Eq. (6). Substituting the expression of ${\widehat{O}}_n$ given by Eq. (8) in the expression of $\mathcal{F}$ given by Eq. (6) leads to a minimization of a polynomial with respect to ${\alpha }_n$. Hence, ${\alpha }_n$ is determined analytically.

In many cases introducing the positivity a priori information of the parameter of interest, herein the density of fluorophores, improves the performance of the inverse scheme [22, 23, 24]. This is realized by retrieving an auxiliary function $\widehat{\xi }$ such that $\widehat{O}={\widehat{\xi }}^2$. The inverse scheme presented above remains unchanged provided that the gradient $g_n$ is replaced by the gradient $g_{n;\xi }$, which is the gradient of the cost functional $\mathcal{F}$ with respect to $\widehat{\xi }$ instead of $\widehat{O}$:

5. NUMERICAL RESULTS AND DISCUSSION

When the sample is illuminated by a homogeneous beam, the recorded intensity I is equal to the density of fluorophores O convoluted by the point spread function P of the microscope. The resolution of the image is given by the full width at half-maximum of P which, for $\mathit{NA}=1.45$, is about $0.4\lambda$. This assertion is illustrated In Figs. 4a, 4b , which display a simulation of a TIRFM experiment. Figure 4a shows a theoretical sample made of randomly scattered beads, while Fig. 4b shows the image that would be recorded on a CCD camera if the sample were illuminated homogeneously.

An insight on the ideal resolution of a given imaging configuration can be obtained by estimating the domain of spatial frequencies of O that is theoretically accessible from $I_l$. The larger this domain, the better the resolution. This estimation is easily done when the illumination is homogeneous. It suffices to Fourier transform the recorded intensity I, $\tilde{I}=\tilde{O}\tilde{P}$, where $\tilde{I}$, $\tilde{O}$, $\tilde{P}$ are the Fourier transforms of I, O, and P, respectively. In an ideal nonnoisy system, $\tilde{O}$ is accessible on the spectral domain S where $\tilde{P}$ is nonnull. With the chosen point spread function, S is a disk of radius $2\mathit{NA}{k}_0$. This low-pass filter gives a resolution of about $0.25\lambda$ for $\mathit{NA}=1.45$. There exist many deconvolution techniques that permit amelioration of the images when the point spread function is known [25]. In the present work, we used the iterative reconstruction algorithm presented in Section 4. Figures 4c, 4d display the density of fluorescence obtained by deconvoluting the CCD image, Fig. 4b, without and with the positivity information, respectively. Hereafter, we will always use the positivity information in the reconstruction procedure.

In the standard standing-wave configuration, Fig. 2b, the Fourier amplitude of the density of fluorophores $\tilde{O}$ can be obtained on the extended Fourier domain $S=\{\mathbf{k},k<2\mathit{NA}{k}_0\}\cup \{\mathbf{k},|\mathbf{k}\pm 2{\mathbf{k}}_{\mathrm{inc}}|<2\mathit{NA}{k}_0\}$ [7, 20], where $2k_{\mathrm{inc}}$ represents the spatial frequency of the light grid illuminating the sample. The higher $k_{\mathrm{inc}}$, the larger S and the better the resolution. With the chosen standing-wave pattern, the accessible spectral domain obtained with structured illumination is almost twice as large as that obtained with homogeneous illumination. Thus, one expects to ameliorate the resolution of the image by a factor of 2.

When the sample is deposited on a grating, one cannot easily infer $\tilde{O}$ from ${\tilde{I}}_l$ as in the standing-wave approach. However, we may still try to estimate the potential resolution of the imager from the analysis of the highest spatial frequencies of the light grid illuminating the sample. The intensity obtained above the grating when the latter is illuminated by one plane wave with ${\theta }_{\mathrm{inc}}=65\mathrm{deg}$ and ${\varphi }_{\mathrm{inc}}=\pi ∕2$ can be approximated by

When the grating is illuminated by two plane waves, the intensity above the substrate is no longer periodic. For ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$ and $\varphi =\pi ∕2$ and a phase difference of ψ between the two waves, it can be approximated by

The successive amelioration of the resolution of the standard TIRFM with the standing-wave, the grating, and the standing-wave-grating configurations are illustrated in Fig. 5 . The sample, made of beads with various interdistances, permits quantification of the resolution improvement brought about by the different configurations. The best resolution is obtained in the standing-wave-grating configuration, which permits one to distinguish two beads that are separated by $\lambda ∕10$. Note that when the grating is illuminated by only one plane wave, some beads are not properly retrieved due to the residual inhomogeneity of the average excitation. This problem disappears if the grating is illuminated by two interfering beams. To assess the ability of the different techniques to retrieve the fluorescence density of more complicated samples, we plot in Fig. 6 the reconstructed density of the random sample of beads depicted in Fig. 4 for the standing-wave, the grating, and the standing-wave-grating configurations. Once again, one observes that using the grating substrate improves the resolution substantially.

6. CONCLUSION

We analyze numerically the resolution improvement brought about by replacing the glass slide of a structured-illumination total-internal-reflection fluorescence microscope by an optimized grating. The grating is illuminated by one plane wave or by two interfering beams at various incident angles. It has been designed so that the periodic intensity pattern obtained at its surface can be translated or rotated by changing the incident angle. Thus, the sample is homogeneously illuminated on average. We develop a versatile inversion algorithm based on a conjugate-gradient technique and accounting for the positivity of the unknowns to retrieve the fluorescence density of the sample from the images. The $160\mathrm{nm}$ period grating optimized in this work permits one to obtain a resolution of about $60\mathrm{nm}$, i.e., a fourfold improvement of resolution in comparison with conventional TIRFM. In theory, even better results could be obtained using gratings with smaller periods.

APPENDIX A: OPTIMIZATION OF THE GRATING

The key issue of the approach presented in this work lies in the optimization of the grating substrate, which should exhibit two important properties. First, the intensity obtained above the grating for one incident angle should be highly contrasted, with high-spatial-frequency features. Second, the intensity averaged over all incident angles should be roughly homogeneous. This second property is essential to avoid the presence of blind spots on the sample. To obtain high-spatial-frequency features, it is convenient to design a grating with a period that is small compared with the wavelength (at least smaller than $\lambda ∕3$ to expect a better resolution than a classical standing-wave illumination). Now, if the periodic permittivity modulation is weak (for example, an engraved glass slide), then, the zero-order amplitude $A_{0,0}$ will be much larger than that of the higher-order terms. As a result, the intensity pattern above the grating will be poorly contrasted. Yet, if the permittivity contrast is high (for example, by placing periodically metallic particles on the glass slide), then changing the incident angles, polarization, or relative phases does not permit, in general, enough modification of the positions where the field is intense or weak, as they are linked to the jumps of permittivity. As a result, the average intensity exhibits blind spots where the fluorescence is never detected. To avoid these two pitfalls, we considered special kinds of gratings that are known as waveguide gratings or resonant gratings. The latter have been studied essentially for their filtering properties [18, 19].

A resonant grating is basically a multilayer that is decorated by a periodic structuration. At the excitation wavelength ${\lambda }^{\prime }$, the multilayer is designed to support a guided or surface mode, described by its wavenumber $k_{\text{mode}}$. If the incident angles and the period of the grating are chosen so that the phase-matching condition,

(14)$$|{\mathbf{k}}_{\mathrm{inc}}+{\mathbf{K}}_{-1,0}|\approx k_{\text{mode}},$$

is satisfied, the incident wave couples to the mode through diffraction with the $(-1,0)$ order. This resonance phenomenon specifically increases the $(-1,0)$ amplitude in the Rayleigh series, Eq. (3). The main issue is then to design a multilayer supporting guided or surface modes with high wavenumbers in order to generate a light grid with the smallest period possible.

In previous work [14, 26, 27], we have considered very thin metallic films supporting high-spatial-frequency surface plasmons (the thinner the film, the higher the plasmon wavenumber). In this paper, following the recent study presented in [28], we have considered a dielectric multilayer supporting guided modes. Indeed, these structures appeared easier to fabricate. The wavenumber of guided modes depends essentially on the index of refraction of the guiding layer. Now, in optics, the refractive indices of the lossless dielectrics that are generally used for waveguiding are smaller than 2.5, which limits $k_{\text{mode}}$ to $2.5k_0^{\prime }$, where $k_0^{\prime }=2\pi ∕{\lambda }^{\prime }$. Hence, to further increase the value of $k_{\text{mode}}$, we have considered a lossy semiconductor, silicon, which presents a high index of refraction with small losses $n_{\mathrm{Si}}=4.+i0.1$ at ${\lambda }^{\prime }=630\mathrm{nm}$. Of course, this material cannot be used for standard waveguiding applications since the mode propagates only over a few micrometers. Yet, it can be used for our application because the resonance phenomenon is strong enough to increase $A_{-1,0}$ at the same level as $A_{0,0}$ with an appropriate design of the periodic nanostructure. The resonant grating that is used in this work is made of one silicon layer of thickness $h=88\mathrm{nm}$ deposited on a glass substrate. This layer supports a TE guided mode with $k_{\text{mode}}\approx 3.5k_0^{\prime }$ at ${\lambda }^{\prime }=630\mathrm{nm}$. It is decorated by periodic holes on a triangular mesh. The period d is chosen so that Eq. (14) is satisfied for ${\theta }_{\mathrm{inc}}=65\mathrm{deg}$ and $\varphi =\pi ∕2$, $d=157\mathrm{nm}$. Note that to excite a TE guided mode in these conditions, the incident plane wave must be s-polarized; that is, the incident electric field is perpendicular to the plane of incidence. The diameter $e=75\mathrm{nm}$ and depth $p=50\mathrm{nm}$ of the holes are optimized to maximize the value of $A_{-1,0}$. To deal with a flat interface, the grating is planarized with a thin layer of resin that fills the holes $n_{\text{resin}}=1.5$. The geometry of the grating is depicted on Fig. 7 .

ACKNOWLEDGMENT

This research is supported by the Agence nationale de la recherche (ANR)-2008-Pnano-P052-36 Sonde Optique Ultra-Résolue pour l’Imagerie de Surface (SOURIS) project.

 

Fig. 1 Schematic view of the TIRFM that is simulated in this work. (a) The fluorophores deposited on the substrate (glass slide: CG) are illuminated by a parallel beam that is totally reflected at the glass/air interface. IO is an inversion objective, and the image of each fluorophore is formed on the CCD camera, through the optical system (IO, L). (b) ${\theta }_{\mathrm{inc}}$ is the incident polar angle and ϕ is the azimuthal angle. ${\mathbf{k}}_{\mathrm{inc}}$ is the tangential component of the incident wave vector.

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Fig. 2 Illumination configurations. (a) Glass slide illuminated homogeneously by a plane wave under total internal reflection. (b) Glass slide illuminated by a standing-wave pattern. The periodic intensity pattern above the substrate is generated by the interference of two plane waves. (c) Grating illuminated by one plane wave. The period of the intensity pattern just above the grating can be much smaller than that obtained in (b). (d) Grating illuminated by two interfering plane waves. The intensity pattern above the grating is not periodic in general.

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Fig. 3 Excitation light patterns generated by the illumination configurations depicted in Fig. 2, (a), (b) Simulation of the field intensity just above the glass slide when the illumination is made of two interfering plane waves with ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$ and relative phase 0 and $2\pi ∕3$ respectively. (c), (d) Simulation of the field intensity just above the grating when the illumination is an s-polarized plane wave with ${\theta }_{\mathrm{inc}}=65\mathrm{deg}$ and azimuthal angle $\varphi =\pi ∕2$ and $\varphi =3\pi ∕2$, respectively. The intensity is calculated with the rigorous Fourier modal method [17]. (e), (f) Same as (c), (d) but the grating is illuminated by two s-polarized plane waves with ${\theta }_{\mathrm{inc}}=\pm 65\mathrm{deg}$, $\varphi =\pi ∕2$, same amplitude and relative phase 0 and π, respectively;

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Fig. 4 Simulation of a TIRFM experiment. (a) Theoretical sample: (beads with a diameter of $20\mathrm{nm}$, randomly scattered on a glass slide). (b) Simulation of the image recorded on the CCD camera using a homogeneous illumination. (c) Deconvolution of (b) with the conjugate-gradient algorithm depicted in Section 4 without assuming the positivity of the density of fluorescence. (d) Same as (c) but the algorithm accounts for the positivity.

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Fig. 5 Illustration of the resolution improvement brought by the grating. (a) Theoretical sample: beads with a diameter of $20\mathrm{nm}$ that are separated by $\lambda ∕2$, $\lambda ∕4$, $\lambda ∕6$, $\lambda ∕8$ from bottom to top on the left side and $\lambda ∕10$, $\lambda ∕14$ from top to bottom on the right side. (b) Simulation of an image recorded on the CCD camera using homogeneous illumination. (c) Normalized density of fluorescence obtained with the iterative inversion procedure accounting for the positivity (Section 4) in the homogeneous configuration; see Fig. 2a. (d) Same as (b) but in the standing-wave configuration; see Fig. 2b. (e) Same as (b) but in the grating configuration; see Fig. 2c. (f) Same as (b) but in the standing-wave grating configuration; see Fig. 2d.

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Fig. 6 Reconstructed fluorescence density of the random sample depicted in Fig. 4. (a) Standing-wave configuration; see Fig. 2b. (b) Grating configuration; see Fig. 2c. (c) Standing-wave grating configuration; see Fig. 2d. (d), (e), and (f) zoom of (a), (b), and (c) respectively.

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Fig. 7 Schematic representation of the grating. (a) Top view. (b) Side view; $d=157\mathrm{nm}$, $e=50\mathrm{nm}$, $h=88\mathrm{nm}$, $l=19\mathrm{nm}$, $p=50\mathrm{nm}$, incident wavelength $\lambda =630\mathrm{nm}$.

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