1. Introduction

High resolution optical microscopy is a vital analytical tool in cell biology as it offers a non-invasive and non-destructive diagnostic technique [1,2]. Aided by optical tagging, high resolution microscopy can be used to image single living cells on a sub-micron scale in order to probe the complex processes that occur within them [3]. Microscopy available for cell-biology has undergone major advances in recent years. Phase contrast microscopy has become a standard technique to provide enhanced contrast whilst structured [4,5] and single-plane illumination [6] offer significant gains in resolution. In addition modern microscopy benefits enormously from the development of high-sensitivity low-cost array detectors and most recently, the exploitation of the nonlinear photo-physics of dyes for resolution well beyond the Abbe diffraction limit [7]. However, a living cell is a constantly changing, three dimensional (3D) object. An ideal microscope system would provide high resolution imaging in real-time and in 3D. Current microscopy techniques are limited in their ability to extract such information. The two main 3D imaging techniques are scanning confocal microscopy and wide-field imaging [1]. 3D imaging in a confocal system is achieved by scanning the laser spot relative to the sample. It gives excellent sectioning properties by rejecting out-of-focus light but scanning limits the imaging speed. Wide-field microscopy gives real-time imaging in 2D but constructing full 3D images requires z-scanning or holographic imaging which, although successful, relies on complicated data analysis [8,9]. An illustrative task is the 3D tracking of a nano-particle, for instance a labeled protein or virus, within a living cell. The 2D co-ordinate, the x-y position, is easily determined, for example using the image “centre of mass” [10]. The challenge is the accurate determination of the z position. Techniques based on utilizing out-of-focus aberrations [11,12] and variations of the standard confocal system [13] have yielded promising results. However, high resolution in all three directions is achieved only in a small region around the focal plane. Furthermore, these techniques do not yield a 3D image of the host cell which may be changing continuously.

2. Three dimensional imaging using a diffraction grating

Our approach to real-time 3D imaging and tracking is to record images from different object-planes simultaneously. We do this by adding an image-relay system to the camera port of a standard microscope, the relay system consisting of a distorted diffraction grating and lens. A diffraction grating with parallel lines confers an order-dependent change of direction to a light beam but does not introduce any order-dependent focusing power. A quadratic distortion of the lines, as in an off-axis Fresnel lens, produces positive focusing power in the positive diffraction orders, negative focusing power in the negative orders and leaves the zeroth order beam unchanged [14]. By exploiting this concept, a lens and a quadratically distorted (QD) grating can be used to image multiple object z-planes simultaneously onto a single image-plane, the camera, Fig. 1c ., as has been demonstrated on a macroscopic length scale [13] with applications also in wavefront sensing [14]. We present here an application on a microscopic length scale: using the distorted grating in an image relay system positioned at the microscope camera port, the separation of the object-planes (determined by the properties of the grating, the magnification of the microscope and the focal length of the relay lens) ranges from zero to a few tens of microns, ideal for applications in cell-biology.

 

Fig. 1 Schematic of the microscope and grating operation. (A) The custom built wide-field 100x fluorescence microscope. Sample illumination is either from an interchangeable source below the sample or for fluorescence studies, fibre-launched excitation light from above. (B) A 1:1 relay system allows the QD grating to be positioned in the telecentric position. (C) Operational schematic of the QD grating. Multiple object-planes, separated by Δz, are simultaneously imaged onto a single image-plane, spatially separated by Δd.

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Our optical system produces three images side by side on the camera, associated with diffraction orders m = −1, 0, and + 1. We demonstrate simultaneous multi-plane imaging in the context of biological microscopy, using both bright-field and fluorescence imaging. We then present a technique, based on image sharpness, for inferring the z-position of a single nano-particle to within a few nanometers from the multiple images. Our approach offers all of the advantages of multi-plane imaging using beam-splitters and multiple cameras [15,16] with the additional advantage of simplicity, low cost and full compatibility with commercial microscopes.

3. Experimental set-up

The grating forms part of a 1 to 1 magnification relay system attached to a custom built, high magnification wide-field microscope, Fig. 1(a). The sample is placed on actuators with on-axis position readout with an accuracy of better than 2 nm. Illumination is from below with either He-Ne laser light (633 nm wavelength) launched via a single mode fiber or with a single LED (~500 nm wavelength) providing 30 nm broadband wide-field illumination. For fluorescence excitation, light is provided from above via the objective lens. A dichroic mirror separates the fluorescence and excitation pathways. A 100x Nikon Superfluor oil immersion objective (NA 1.3) in combination with a 200 mm tube lens creates a magnified image in the microscope image-plane. An aperture controls the field of view and is adjusted to prevent the images on the camera from overlapping. The relay system consists of two identical 400 mm focal length lenses separated by 50 mm and acts as a single compound lens of effective focal length f₁ = 213 ± 5 mm. The relay system projects the image from the microscope image-plane onto the CCD camera, Fig. 1(b). To avoid order-dependent magnification the grating is positioned in the telecentric position [17], one focal length away from the relay lens, 221 mm ± 10 mm from the first principal plane of the compound lens. The telecentricity is experimentally verified to be within 1% with an overall magnification of the microscope and relay lens system of 105 ± 1 in all orders. In the telecentric position the axial separation of the object-planes in the sample is given by Δz_m = f₁ ² / M²f_m where M is the magnification, f₁ is the focal length of the relay lens and f_m is the focal power in the m^th diffraction order [17].

4. Microscope characterization

The performance of the microscope was characterized by using a single nano-hole. The nano-holes are 200 nm diameter apertures in an opaque metal mask. Latex spheres (200 nm diameter) are spin coated onto a 0.2 mm thick glass cover slip and coated with 5 nm of NiCr followed by 75 nm of Al and a further 8 nm of NiCr. Immersion in toluene expands the spheres which consequently fall out of the mask to leave the nano-holes. The NiCr/Al/NiCr layer provides a robust metal coating resistant to oxidation and damage with optical attenuation of 5 orders of magnitude. The average separation of the nano-holes is several tens of microns. The nano-holes are positioned close to the focal-plane of the objective lens and back-illuminated using 633 nm laser light. A single nano-hole is an approximate point source and mimics a single molecule emitter or nano-scatterer. The QD grating used here is a binary phase grating with features etched into silica and is designed to be approximately intensity-matched in the −1, 0 and + 1 orders for 633nm light (etch depth of 450 nm). It has a central period 50 μm, and an effective focal length of + (-) 4.94 m in the + (−1) order for 633 nm incident light.

At each z position a single camera-frame gives three images corresponding to m = −1, 0 and + 1, Fig. 2 . Figure 2(a) shows the evolution of these images as the nano-hole is moved through the focal volume in the z direction. For m = 0, the image is sharpest at zero defocus, z = 0. This position corresponds to the objective focal plane. At positive defocus, z>0, this image blurs, developing pronounced rings. At negative defocus, z<0, the image again blurs but without the rings. The image asymmetry in z arises from spherical aberrations in the objective lens, and is well known. The novel feature in Fig. 2(a) is the behavior of the diffracted images. For m = −1, the image is blurred at z = 0 but sharp at z = −1.12 μm. Conversely, for m = + 1, a sharp image is formed at z = + 1.12 μm. In fact, the series of images for m = −1, 0 and + 1, are essentially identical but offset in z. This demonstrates the multi-plane capability of the microscope. The plane separation, 1.12 μm, agrees reasonably with the expected value, 0.9 μm, based on the properties of the grating and geometrical optics [17].

 

Fig. 2 (A) A single nano-hole imaged as a function of defocus position using the QD grating system and 633 nm illumination. Each image corresponds to an area of 6x6 μm² (0.63x0.63 mm²) in the object (image) plane. (B), (C) and (D) mark the focal points for the −1, 0th and + 1 orders, respectively. (E), (F) and (G) show cross-sectional slices through each of these respective images along with Gaussian fits to the data.

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The lateral resolution was assessed by cross-sections through the images in Fig. 2. Cross-sectional slices through the three sharpest images from each order are shown in Fig. 2(e-g) along with Gaussian fits to the data. For m = 0 the full width of the intensity profile is 242 nm(λ/2.63), equal to the full width recorded without the QD grating in the relay system. All three images show similar resolution. The + 1 and −1 images impinge on the camera at an angle. However, this angle is very small, less than 0.1 degrees in our system, and has no adverse effect on the image quality [14], as shown by the symmetry in the curves in Figs. 2(e) and 2(g). The conclusion is that the QD diffraction grating preserves the high image quality of the microscope in all orders.

5. Single cell imaging

Multi-plane imaging is demonstrated in a biological context using a single leaf protoplast (a single plant cell with the cell wall removed) from the model plant Arabidopsis thaliana. The Arabidopsis line used expresses Green Fluorescent Protein (GFP) which is targeted to the mitochondria [18]. Figure 3(a) shows a single Arabidopsis cell in wide-field mode illuminated from below with 510 nm light. The three images correspond to object-planes at z = −0.9, 0.0 and + 0.9 μm. Distinct features can be made out in each order, a result strongly highlighted when imaged in fluorescent mode, Fig. 3(b). The three images show the differing distributions of mitochondria at different focal depths. Similar results have also been obtained with single He-La cells expressing mitochondrial-targeted GFP and single Chinese hamster ovary cells expressing mitochondrial-targeted DsRed fluorescent protein.

 

Fig. 3 (A) A single Arabidopsis protoplast imaged in bright field using 510 nm wavelength LED illumination. (B) Artificially colored image showing 510 nm wavelength fluorescence from mitochondrial-targeted GFP. The illumination is provided by a 470 nm wavelength laser. The grating gives an object-plane separation of 0.9 μm at 510 nm wavelength.

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6. Particle tracking

The multi-plane imaging also allows 3D nano-particle tracking. A nano-particle moving in the z direction gives a sharp image first in one order, and then the next, and so on. This allows an unambiguous and high precision determination of the particle’s z position. The challenge is to find a simple and quick method to infer this position from the images which, as shown in Fig. 2, depend sensitively on the out-of-focus aberrations in the optical system. We meet this challenge by characterizing each of the multiple images in a single camera frame by its ‘image sharpness’ and inferring the z-position from the multiple sharpness values using a maximum likelihood estimation (MLE).

The image sharpness (s) is the integral of the image intensity squared and gives a single measure of image quality [19]. Sharpness uses the entire image so that all the data recorded by the camera contributes. Figure 4(a) shows the image sharpness calculated from a set of 70 images with 112 nm axial object-plane spacing. We find that the sharpness as a function of z forms a simple, bell-shaped curve for each diffraction order, with minor features dependent on the aberrations. For each order, the sharpness peaks at the z-values where the cross sectional image width is minimum, Fig. 2. Each curve is close to symmetric about the maximum. The unequal peak sharpness values are due to spherical aberration in the objective and to more efficient flux collection from planes closer to the objective. In fact a key feature of our approach is that, as the calibration data is taken experimentally as discussed below, our technique is insensitive to the exact nature and details of the aberrations within the system.

 

Fig. 4 (A) The image sharpness from each diffraction order as a function of defocus. At each defocus position 50 separate images (exposure time 0.5 ms) were taken at 100 msec intervals. Plotted is the average sharpness. (B) The standard deviation of the maximum likelihood estimate analysis of the sharpness data from 50 images at each z position.

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A single camera frame leads to a unique solution for the z coordinate. For each image in each diffraction order we calculate the sharpness. For the m^th diffraction order, p(s_m|z) is the conditional probability density function for the sharpness, s_m, given the object position z. For a measured sharpness value, p(s_m|z) produces a curve with two Gaussian peaks, one arising from the solution for z on the “left” of the bell-shaped sharpness function, one from the “right”. With the QD grating, p(s_m|z) for all orders can be combined such that the likelihood for the measurement set {s_-1, s₀, s₊₁} obtained from each camera frame is given by,L({s_-1, s₀,s₊₁}|z) = p(s_-1|z)p(s₀|z)p(s₊₁|z).

The likelihood function produces a single, well-defined maximum signifying an unambiguous solution for z. We have tested this for z values between −4 μm and + 4 μm and find a unique maximum likelihood throughout this range. We quantify the error in the position resolution by recording 50 frames at a fixed z position, hence providing 50 MLE z positions. The average of the MLE z positions is compared to the z position recorded from the scanner, providing a calibration for the MLE z position to within a 1 nm systematic error. The standard deviation of these MLE z positions reaches a minimum of about 6 nm mid-way between in-focus positions, rising to 50 nm at z = + −4 μm, and averages 11.85 nm in the range between the + 1 and −1 focal planes, Fig. 4(b). This standard deviation includes all the random errors in the procedure and is dominated in our case by environmental noise. The random error in locating the maximum in L(z) from a single shot measurement is much smaller, sub-nm for the range of data shown in Fig. 4.

8. Compatibility with a commercial microscope

To test the compatibility of our multi-plane imaging system with a commercial microscope, the QD grating relay system was connected to the output port of an Olympus IX71 microscope. This has a more complicated optical arrangement than our custom built system, Fig. 1, and in fact the exact details are unknown to us. Nevertheless, we have succeeded in reproducing all the main results: multi-plane imaging with an object-plane separation of 1 μm; ideal images in all three diffraction orders; and simple sharpness functions. When applying the MLE we reproduce the results of the custom built system, including the tracking accuracy. We have demonstrated the imaging method in fluorescence microscopy, bright-field, dark-field, phase-contrast and differential interference contrast (DIC) imaging modes.

9. Outlook

Multi-plane imaging using a QD diffraction grating can be extended in a number of simple ways. First, a set of gratings each with a different focusing power can provide a variety of object-plane separations. A slider assembly, or filter wheel, can be used to change gratings. Alternatively, the gratings can be formed digitally using a spatial phase modulator and then changed by simply reprogramming such a device. Second, crossed gratings can be used to provide more image-planes (9, 27 etc) and produce a 2D array of images, utilizing the maximum area of typical imaging cameras. Obviously, there is a distribution of the available light into each order. However, compared to a traditional z-scan, bleaching and motion-blur are uniform throughout the specimen-volume imaged and, through efficient distribution of flux between the diffraction orders, the measurement time to acquire a z-stack with given signal to noise is unchanged. Third, chromatic smearing introduced by the QD grating can be eliminated by pre-dispersing the incident beam, eliminating the necessity of working with a narrowband light source [20]. Finally, video-rate nano-tracking with 1 nm position resolution with our concept is well within reach provided adequate photon flux.