1. Introduction

The ability to obtain two dimensional optical sections of three dimensional samples is of great interest for 3D imaging. So far, 2D optical sections have been obtained using mainly confocal microscopes [1]. Although this technique offers great advantages, it is limited by the fact that 2D scanning of the sample is necessary. This limitation is partially bypassed by recent technical refinements including the spinning disk confocal [2] and the “line scan” confocal [3]. These techniques enhance the speed of acquisition, but have their own limitations. In both cases, these techniques are quite hard to implement optically. In the case of the spinning disk, the microscope sectioning properties are hardly tunable.

An other type of microscopy offering optical sectioning has been introduced almost ten years ago: “the structured illumination microscopy (SIM)”. Firstly suggested by Lukosz and Marchand [4] to improve the lateral resolution of an optical system, this technique has been used by Neil et al. [5] to improve the axial resolution of a microscope. It is a wide field based illumination technique and is therefore scanning free. The idea is to project a stripe pattern on the sample observed. This stripe pattern introduces an artificial high frequency spatial modulation of the specimen of interest. Because high spatial modulations attenuate very rapidly when we move away from the focus, it is possible to isolate, using simple image analysis, the part of the sample that is modulated and thus in focus, from the background that is not modulated and thus out of focus. This artificial spatial modulation, coupled to image analysis enables wide field optical sectioning of the sample. The strength of the sectioning depends on the pitch and the contrast of the stripe. The image analysis is done in such a way so that the stripe pattern is removed from the final reconstructed optical section.

So far, the structured illumination technique principle has been described, but no quantitative theoretical or experimental analysis of the the resolution increase along the optical axis, or the influence of the signal/background ratio has been done. Because this technique is very simple to implement on a regular microscope and because its performances depend greatly on the stripe pattern characteristics (pitch and contrast), we have carefully evaluated these parameters both theoretically and experimentally.

We have also added major modification to the original set up. The stripe pattern modulation described so far is done in a step wise manner: the stripe pattern is moved by small increments and images are acquired in between the grid displacements. Although this approach works fine when low frequency movements are applied, it becomes imprecise when images are acquired at video rate: the piezoelectric used to move the grid in between the incremental moves needs time to stabilize before an image is acquired. This kind of displacement is also sensitive do drift and hysteresis. This drawback reduces the practical rate of sectional cut acquisition. We show that this problem can be bypassed using a sinusoidal modulation of the stripe pattern coupled to the combination of four images acquired during one modulation period. As we will see, using this technique, four images per modulation period are enough to obtain wide field optical sections of the sample. Thanks to this refinement, high temporal modulation frequencies can be used for the movement of the stripe pattern and optical sections can be obtained at frequencies close to video rate. We investigate the thickness of the optical cuts as a function of the stripe pattern pitch used. We show that when this pitch is correctly chosen the axial resolution can be increased by a factor 1.5 compared to the standard wide-field microscope. We also study the contrast of the sectioned images when the amount of out of focus fluorescence increases to evaluate the minimum detectable signal. We also show that the contrast of the stripe pattern has a great influence. Finally, we compare our system to commercial solutions: confocal microscope, spinning disk and structured illumination microscope by imaging fluorescent 100nm beads under similar conditions.

2. Structured illumination microscopy

2.1. The discrete modulation approach

As mentioned above, optical sectioning via structured illumination was first demonstrated in 1997 by Neil et al. [5]. The idea is to project a stripe pattern on the sample of interest. This projection produces an additional spatial modulation at a frequency (υ_g) on the image detected. When well chosen, this spatial frequency attenuates rapidly with defocus. As presented in ref [5], optical sections can be obtained by linear combination of three images obtained for different positions of the stripe pattern on the sample. The intensity in the image plane can be written:

where I_C is the intensity of the signal that is not modulated by the projection of the stripe pattern, it corresponds mainly to the out of focus signal. I_S is the intensity of the signal that is modulated by the stripe pattern, it corresponds mainly to the in focus part of the signal. υ_g is the spatial frequency of the grating in the image plane. φ the spatial phase of the grid pattern. Optical sections S of the sample can be obtained combining three images I ₁, I ₂, I ₃ obtained respectively for $\phi =0,\frac{\pi }{3},\frac{2\pi }{3}$ as follow:

which is equivalent to:

This method has been applied both for reflexion and fluorescence microscopy [6]. The optical cuts using this method are comparable to the ones obtained with a classical confocal microscope. The main disadvantage of this restoration process is its sensitivity to the stripe pattern phase shift. When the grid is not shifted of exactly one third of period, a residual modulation can be observed on the calculated section requiring additional image processing [7]. This sensitivity becomes a strong problem when working at high frequencies as the piezo actuator needs few milliseconds to stabilize in a precise position. We propose here a new way to restore the optical section using a grid sinusoidal modulation less sensitive to drift.

2.2. Sinusoidal modulation and image restoration

We choose to modulate the grid position with a sinusoidal motion around its average position. In the image plane, if the stripe pattern is perpendicular to the x axis, and its position modulation performed along the x axis, equation 1 becomes (we discard the dependence on x and y to simplify the notation) :

X and T are respectively the amplitude and period of the gridmotion. ψ is the phase between the grid motion and the image acquisition. Because of the continuous grid modulation, the process presented above to obtain an optical section is not valid any more. In our case, however, I_S can be computed using a multiplexed lock-in detection method similar to the one used in phase shifting interferometry [8]. Four images I_P with p=1,2,3,4 are acquired by integrating I(x,y,t) during one modulation period as follows:

I_P(x,y) can be computed using a development of I(t) with Bessel functions of the first kind J_n. Details of these calculations can be found in [8]. Using two linear combinations of these four images:

where

we can easily compute the amplitude of the signal I_S. Indeed, as it has been shown in [8], when 2πυ_gX=2.45 and φ=0.98, Γ_S=Γ_C=Γ=0.4. Using these values, we find that:

Using equ. 6, the calculated section S can be expressed as a function of the four images taken during one modulation period :

It thus appears that even in the case of a sinusoidal modulation of the grid position, four images acquired during the modulation period are enough to provide, using multiplexed locking detection, wide field optical sections of the sample. Even in the case of high modulation frequencies, this reconstruction process appears to be very stable and no residual modulation have been observed in the optically sectioned image.

2.3. How does structured illumination affects the Point Spread Function and the Optical Transfer Function ?

The Optical Transfer Function (OTF) of an epifluorescence microscope has a torus-like shape with a greater extend along the lateral spatial frequencies axis (k_x,k_y) than along the axial spatial frequency axis (k_z) [9]. A typical OTF for an epifluorescent set up is represented Fig. 2, top-right. The so called missing cone along the axial axis is clearly visible. This implies that in an epifluorescent microscope, the resolution along the optical axis will be lower than the one along the transverse axis and that some spatial frequencies along k_z cannot be transmitted through the objective (that would be the case for a thin fluorescent film which contains only spatial frequencies along k_z). Interestingly, as we will see, the OTF of an SI is significantly differerent than the one of an epifluorescent microscope. It does not features any missing cone, is larger along k_z, and thus provides improved axial resolution, without any loss in the lateral resolution.

We elected to work with optical coordinates (ρ_x,ρ_y,u) and normalized spatial frequencies (υ′_x,υ′_y) which are related to the actual coordinates (x,y,z) and spatial frequencies (υ_x,υ_y) by [10]:

where $k=\frac{2\pi }{\lambda }$ and NA=nsinα, α being the semi-aperture angle and n the refractive index of the immersion media. The axial response of an SIM when the sample is a thin fluorescent film can be evaluated with [5, 11]:

We can plot the axial response as a function of the normalized spatial frequency of the grid to find that the best sectioning properties are obtained for υ′_g=1 (see figure 1)

 

Fig. 1. FWHM of the axial response in optical units as a function of the normalized spatial frequency of the grid. This shows that the narrowest axial response is achieved for a normalized spatial frequency equal to 1

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We want to evaluate the PSF of the SIM. We calculated the image of a fluorescent punctual source placed in the object focal plane of the objective. The grid G(ρx0,ρy0) is projected on to the fluorescent sample f(ρx1,ρy1). The grid is illuminated incoherently so the intensity in the object plane can be written:

Where h_exc is the amplitude excitation PSF at the wavelength λ_exc. In the plane of the camera, the total intensity can be written:

Where h_em is the amplitude emission PSF at the wavelength λ_em. The object is a punctual source so f(ρx1,ρy1)=δ(ρx1,ρy1) which leads to:

Using equation (18) we can evaluate the sectioned image intensity by combining four images where the expression of the grid G takes into account its displacment. We found that using equation (2) greatly simplifies calculations without changing the results in terms of resolution. The grid can be depicted:

where p=1,2,3. We obtain for the optically sectioned intensity:

We can introduce the pupil function P(υ_x,υ_y) through the Fourier transform of the amplitude point spread function:

We can write the point spread function of the structured illumination microscope as the product between the point spread function of the conventional microscope by the optical transfer function. We extend the calculation to the optical axis using the Stokseth[11] approximation so we finally obtain:

The second term does not show any transversal dependence which indicates that the lateral resolution remains unchanged. On the contrary it has strong longitudinal dependence so the axial resolution can be increased. When the normalized spatial frequency is set to 0, which corresponds to removing the grid, we obtain the conventional PSF.

We calculated the Fourier transform of the SI PSF to obtain the OTF in the case υ′_g=1. Results are presented on figure 2.

 

Fig. 2. PSF and OTF for an epifluorescence microscope and for structured illumination: the OTF of structured illumination microscope (bottom right) is enlarged compared to the OTF in the epifluorescence case (top right). This is consistent with the PSF of the structured illumination case that is narrower along the optical axis than that of the epifluorescence case

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2.4. Consequences: improved sectioning ability and resolution along the axis

The enlargement of the OTF along the axial spatial frequencies shows that the resolution is improved along the optical axis. The axial resolution improvement depends both on the imaging lens and on the grid normalized spatial frequency υ′_g. In order to obtain the highest resolution improvement, a stripe pattern with a normalized spatial frequency close to 1 is necessary (see Fig. 1). If such a condition is fulfilled, the axial resolution can be improved up to a factor 1.5. As opposed to the technique developed by M. Gustafsson [12, 13]. the SI set up studied here does not provide any improvement of the lateral resolution.

When υ′_g is lower (or higher) than 1, the axial resolution may not be improved. Nevertheless, the sectioning property of the microscope is still present and the background fluorescence is removed. The optical thickness achieved depends on the pitch of the grid and the numerical aperture of the objective. As an example, table 1 presents achievable thickness with different pitches and objectives.

2.5. Signal, noise and sensitivity

We have seen that the sectioning strength of the optical cut depends both on the imaging lens and on the stripe pattern spatial frequency. There is an other parameter yet that influences greatly the quality of the optical cut obtained : the stripe pattern contrast 0<m<1. On an optical set up, a contrast m=1 is impossible to achieve because the imaging lens strongly attenuates high spatial frequencies. With our setup we were able to obtain a contrast as high as 0.55. When a grid is placed in the field diaphragm of a commercial microscope, the contrast obtained is usually around 0.2–0.3. The contrast is important because it determines the modulation depth and thus the image intensity of the optical section. To grasp the influence of the stripe pattern contrast, we studied how m affects the detection threshold of an optical section. The higher the contrast, the better the detection threshold. Let S be an optical section of a sample

Table 1. Axial response in microns and corresponding υ′_g: this table summarize the achievable thicknesses evaluated as the FWHM of the axial response (given in microns) for different grid pitches (given in line pair per millimeter) for different objectives and grids. We also indicate the corresponding normalized spatial frequency of the grid in the plane of the specimen υ′_g which was calculated accordingly to our experimental setup. The wavelength λ was set to 0.6µm.

View Table

computed using equation (10). For simplicity we suppose here that the sample is homogeneous, so that every pixel of the camera detects the same signal modulo the noise. Let 〈S〉 be the image average (here the average is on all the pixels, since the sample is taken homogeneous), and ${\sigma }_S={〈S^2-〈S^2〉〉}^{\frac{1}{2}}$ the standard deviation of the image (the average here again is made over all the pixels). We define the optical section contrast as C=〈S〉/σ_S. We impose that an optical section is meaningfully detected if C>2. To compute C as a function of m, we begin with equation 1 which can be written as a function of m as:

Using a signal described by equation (23), we simulated four images on a CCD camera recorded as the stripe pattern moves over a modulation period. The simulation of the acquisition is done as described in section 2.2. 〈S〉 is evaluated by combining images without noise using equation (11). σ_S is evaluated from the combination of four images where the shot noise has been added to each image. S is then computed using as a function of γ=I_S/I_C for m=0.1,0.5,1. As it appears figure 3, the optical section contrast depends greatly both on I_S/I_C and on m. For m=0.5, which is close to the highest experimental value that we managed to obtained, we found that an optical section can be obtained (C>2) if γ>5×10^-2. Consequently, even if the background fluorescence is 20 times higher than the modulated signal, an optical cut can still be obtained. In the case of fluorescence microscopy for biology, samples are usually specifically tagged. The consequence is that only few elements of the sample emit fluorescent light so the background intensity is low. Under this conditions, structured illumination can provide optical cuts with a very good contrast.

2.6. Experimental setup

We use a home made microscope (c.f. figure 4) with a Xenon monochromator (Cairn), a kohler illumination with a transmission grid (G, Edmund optics) placed in a plane conjugated with the field diaphragm (FD). The grid is placed in the focal plane of the objective 1 (O ₁,Olympus x5/0.1) thus projected at the infinite. It is then focused by the second objective (O ₂, Olympus oil immersion objective x100/1.4) in its focal plane. This way the part of the object that are in focus have the image of the grid superimposed. The transverse position of the grid is modulated with a pieazoactuator (Piezo jena) controlled by a waveform generator(Agilent33220A). The frame acquisition by the CCD detector (PixelFly,PCO) is triggered by a second waveform generator locked in phase and synchronized with the motion of the grating. As described in the theory section, four frames are recorded by modulation period and the computed image S is displayed by a home written software.

 

Fig. 3. evolution of the optical section contrast C as a function of γ for different values of the contrast m

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Fig. 4. experimental setup: the use of a microscope objective in the illumination ensures a good contrast of the image of the grid

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Even if this setup can be implemented on a commercial setup, we found that the contrast of the projected grid could be improved by changing the lenses in the illumination path. We choose to use an objective instead of a simple lens to ensure a good contrast of the grid pattern in the focal plane of the second objective. We show that optimizing the contrast of the grid also optimizes the SNR of the final image. We use three different grid pitches (20lp/mm, 40lp/mm, 80lp/mm) depending on the objective used. Experiments were conducted with the x100/1.4 immersion objective and we elected to work with the 80lp/mm grid. In the image, the normalized spatial frequency of the grid is 0.69 which should lead to an improvement of the axial resolution up to 370nm. The contrast of the image of the grid was measured on a plane mirror and was found to equal 0.55.

3. Comparison with other techniques

In order to have an idea of the performance of our system, we compared its axial resolution to other commercial solutions: confocal microscope, spinning disk microscope and commercial structured illumination microscope. The different techniques were compared by imaging 100nm fluorescent particles (molecular probes 580/605) dried directly on the coverslip to minimize aberrations [14]. Stacks were acquired with a z step of 50 and 100nm using the same NA 1.4 oil immersion objective (Olympus) for each experiment. The sectioning ability was evaluated by the FWHM of the axial response. Images acquisition for each setup is detailed in Appendix. Results are presented in the case of our structured illumination setup in figure 5.

These results suggest that the best resolution can be achieved with structured illumination microscopy. In the spinning disk confocal microscope, the thickness of the optical cut can’t be tuned (the aperture of the pinhole is fixed) and a thickness of 800 nm is announced by the manufacturer which is in good agreement with the results obtained. In the case of commercial structured illumination, results are not surprising. The pitch of the grid is not optimized to increase axial resolution but mainly to remove out of focus fluorescence. Concerning the confocal microscope, the pinhole diameter was set to 80µm which is equivalent to 0.47 Airy units. Under this condition, the pinhole is small enough to increase resolution along the axis.

 

Fig. 5. Experimental comparison of the wide-field axial PSF and the Structured illumination axial PSF

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4. Conclusion

We demonstrated that the use of structured illumination removes out of focus fluorescence and allow depth discrimination. Indeed, we showed that it can lead to an improvement of axial resolution up to a factor 1.5 compared to the wide field microscope. A great advantaged of this technique is that it can be used either in fluorescence or in reflexion just by changing the dichroic beamsplitter by a semi reflecting plate. The setup is also easily adaptable with minor modifications to a standard epifluorescence microscope and lots of different kind of excitation source can be used (laser, spectral lamp). Because of its sensitivity, this technique may not be used for depth imaging as the background fluorescence increases and the contrast of the grid decreases. In such cases, confocal microscopy offers a better solution. Nevertheless, for thick sample like cells cultures, structured illumination is a simple alternative to more complex setup and offers high resolution possibilities.

The use of the sinusoidal modulation allows higher modulation frequencies than a saw tooth modulation, the response of the piezo actuator is not distorted so the grid is well shifted and the reconstruction algorithm still works properly. Images can be processed in real time, the total exposure time to acquire one optical section was set to 200 ms allowing fast tomographic imaging. One major drawback of this technique is that the sample should not move during the acquisition of the four frames because this would led to the apparition of artifacts. Nevertheless, the same problem occurs for scanning techniques where the point of an image are acquired sequentially, that is to say not at the same moment. To conclude structured illumination offers more than a cheap alternative to other techniques like confocal microscopy. The achievable resolution are the same as to that offered by confocal microscopy. This techniques has the advantage to work on a wide-field microscope with only few modifications as it does not need a scanning device. It can be used either in reflexion or in fluorescence with all kind of light source.

This work has been supported by the Ministère de la Recherche, the Centre National de la Recherche Scientifique and the Société des Amis de l’Ecole Supérieure de Physique et de Chimie Industrielles. Authors are grateful to Jean-Christophe Olivo-Marin (Analyse d’images Quantitative, Institut Pasteur), Emmanuelle Perret and Pascal Roux (PFID, Institut Pasteur) for helpful discussions and for providing access to confocal, spinning disk and Apotome microscopes.