1. Introduction

Optical microscopy utilizes various phase- and amplitude contrast enhancement techniques for intrinsically low-contrast samples. An interesting more recent technique, which offers the possibility to generate orientation-independent edge enhancement or to display the “phase-relief” of a sample, is called “spiral phase contrast” (SPC) microscopy [1, 2] or Hilbert-microscopy [3, 4]. There, a helically shaped phase profile is imposed onto the image wave in a Fourier plane of the optical path. This corresponds to a convolution of the image with a Laguerre-Gaussian (i.e. “doughnut”) point spread function (PSF), and results in strong isotropic edge-enhancement of amplitude and phase samples [5, 6].

Typically the transmission function T of the filter corresponds to the phase distribution of a Laguerre-Gaussian (LG) beam with helical charge (or helical index) l of l = 1, i.e. T = exp(ilφ), where φ is the polar angle measured from the center of the spiral phase plate [7]. In general no particular attention is paid to the sign of the helical index l = ± 1 (i.e. the sense of rotation, which can be either left or a right-handed), since normally it does not influence the appearance of the processed images.

The situation changes, however, if the sample contains phase singularities which are superpositions of Laguerre-Gaussian vortex dislocations. Let us consider - as the most basic case - a phase sample, which locally contains a LG phase vortex with a helical index of l = + 1. This situation is depicted in Fig. 1 where the test sample consists itself of a spiral phase plate (SPP).

 

Fig. 1 Imaging of a spiral phase plate (SPP) used as a test sample. The SPP converts an incoming Gaussian illumination beam into a LG_{0 + 1} mode. After diffraction by a binary spiral phase hologram the beam is back-transformed into a Gaussian beam in one of the diffraction orders (focused spot), and transformed into a LG_{0 + 2} beam (focused doughnut ring) in the other order.

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Here we assume that this sample object is illuminated by a plane wave (indicated by a set of normally incident plane phase fronts). The wave transmitted by the SPP is in a LG mode with helical index of 1, i.e. LG_{0 + 1} (indicated by a propelling wave front). The wave is then diffracted by a binary spiral phase hologram (corresponding to an off-axis phase hologram of a first order spiral phase plate, which consists basically of a phase diffraction grating with a fork-like dislocation at its center). Diffraction of the incoming wave into the two conjugate first and minus first diffraction orders leads to a convolution of the incoming wave with two spiral phase Fourier filters with helical indices of l = + 1, and l = −1, respectively [8, 9]. Thus the outgoing wave obtains a LG phase profile with helical charge l = + 2 (corresponding to a 2nd order doughnut), and with l = 0 (corresponding to a Gaussian) in the two conjugate diffraction orders [10, 11]. In the image of the sample the phase singularity at the center of the local vortex then appears in one diffraction order as a focused doughnut ring with a dark center and in the other diffraction order as a focused Gaussian spot with a bright center, respectively.

The model can be generalized to the imaging of arbitrary phase singularities, which can always be mathematically described as a superposition of LG modes with various helical indices. Since the LG modes form a complete basis [12], the model can be generalized to the mapping of arbitrary phase topographies with various superpositions of LG modes: A local singularity is chiral (i.e. non-symmetric with respect to a rotary inversion), if the coefficients of LG modes with opposite signs of their helical indices are different. Using a spiral phase Fourier filter with helical index l = + 1 as a filter kernel results in a corresponding increase of + 1 of all helical indices of the LG modes contained in the image, resulting in an increased helical charge, i.e. an increased absolute value of the helical index, for positive (left-handed) modes, and a decreased helical charge for negative (right handed) modes. The situation is reversed if one uses a spiral phase filter of opposite sign. Considering that the diameter of a focused doughnut ring in the image plane grows monotonically with the absolute value of its helical index [13], one can conclude that all chiral singularities will show a difference when filtered with spiral phase plates of opposite helical indices. Subtracting these two images then results in a map of the chiral singularities in a sample, where the rotational sense can be extracted by analyzing the (positive or negative) signs of the central regions of the structures in the difference image.

Note that unlike other typical applications of SPC microscopy, the goal of the present approach is not to obtain an edge enhancement, nor to produce a phase topography map of the sample, but to detect singular points in the sample, i.e. the positions and the rotational directions of screw dislocations, which can act as a kind of “fingerprint” of a specimen.

As an application of the method we visualize naturally occurring phase singularities at the center of macro-steps in mineral samples, which are formed by the bunching of elemental steps of atomic layers [14–16]. The corresponding phase singularities range from distinct spiral steps with a wide step separation at the center to conical or polygonal appearing pyramids (growth hillocks) in the case of narrow step separations [14]. With conventional standard methods such as phase shift interferometry [17], atomic force microscopy [18], and others, it has been already demonstrated that muscovite or kaolin minerals show spiral growth mechanisms [14], and phase singularities are located at the growth center of these spiral steps. Here we demonstrate that spiral phase filtering with opposite filter orientations can be used as an easy-to-implement alternative to visualize these phase singularities. Note that screw dislocations may be also investigated by means of polarizations microscopy [19], which is, however, limited to the case of birefringent samples, whereas the spiral phase filtering method is also applicable for non-birefringent specimen.

2. Experiments

In our measurements we employed the setup shown in Fig. 2. It basically corresponds to a light microscope, with a spatial light modulator (SLM) located in a Fourier plane of the optical path, which displays an off-axis version of a binary spiral phase filter.

 

Fig. 2 Schematic sketch of the setup (not to scale).

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For illumination we use light from a HeNe-laser (wavelength 633 nm) which is coupled into a multimode acrylic fiber (0.4 mm core diameter) by using a focusing lens (L1, f₁ = 30 mm, the diameter of 1” applies to all other lenses implemented in the setup). Behind the fiber the beam passes through a rotating diffuser (0.5° diffusing angle) to avoid disturbing static speckle patterns by averaging over time-varying speckle fields. This, together with the relatively large thickness (of nearly 1 cm) of the phase plate, prevents any Fabry-Perot effects due to the spiral phase plate [20]. The beam is then collimated by another lens (L2, f₂ = 30 mm) and homogeneously illuminates our transparent sample.

Behind the sample the light is collected by a microscope objective (10x magnification, Reichert, Plan 10, NA = 0.2 or 20x magnification, Olympus, UPlan FL N, NA = 0.5). A set of relay lenses (L3 and L4, with focal lengths of f₃ = f₄ = 200 mm) images the rear focal plane of the objective (corresponding to a Fourier plane) onto the surface of a reflective phase-only SLM (HOLOEYE Pluto, 1920 x 1080 pixels, pixel size 8 x 8 µm²). Between the two lenses (L3, L4), at the intermediate image plane, an aperture is inserted which prevents the different diffraction orders from overlapping in the camera plane, thus also reducing the available field of view.

Due to the fact that there is no clipping of Fourier components as the beam passes all optical components the resolution is determined only by the NA of the objective, i.e.

In order to demonstrate the basic effect, namely the identification of an exemplary phase singularity, we use a commercially obtainable spiral phase plate (first order, optimized for a wavelength of 700 nm) as a test sample. The glass made spiral phase plate is placed directly on the microscope stage in the setup depicted in Fig. 2.

First a simulation of the experimental results is presented, as shown in the first row of Fig. 3. Figure 3(a) displays the simulated spiral phase test sample as a gray level image (i.e. gray levels indicate phase values in the range between 0 and 2π). In order to simulate the left- or right-handed spiral phase filtered image of such a structure, the complex sample image is numerically convolved with the corresponding point-spread functions of left- and right handed spiral phase filters, which are also numerically calculated. The simulations basically comprise a point wise multiplication of the Fourier-transformed intensity image of the sample with either a left- or a right-handed phase vortex, followed by an inverse Fourier transform (for detail see [1]). The results of the two conjugate filter operations are shown in Figs. 3(b) and 3(c), respectively. Subtraction of the two images leads to the difference image displayed in Fig. 3(d), which highlights the dislocation in the center of the simulated vortex, as expected.

 

Fig. 3 Analyzing the handedness of a spiral phase plate used as a sample object. The first row shows the results of a numerical simulation; (a): simulated spiral phase sample: gray levels correspond to phase values; (b): left-handed spiral phase filtered image; (c): right-handed spiral phase filtered image; (d) calculated difference between (c) and (b). The images in the second row (e-h) show the corresponding results of the experimentally recorded bright field image of the spiral phase plate bounded by the field-of-view limiting aperture visible in (e). For reasons of greater clarity pictures 3(f-g) are false color images. Intensities are given in arbitrary units, and pictures (b, d) and (f, h) are normalized to the maximum intensity of pictures (c) and (d), respectively.

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The corresponding results of the experimentally recorded images are shown in the second row of Fig. 3. Figure 3(e) shows the center of the SPP test sample (bounded by the aperture diaphragm) without being filtered, i.e. here the SLM is just used as a plane mirror, and the image appears in the so-called zero diffraction order. As expected for the corresponding bright field image, the phase profile of the SPP sample is only marginally visible. Figure 3(f) shows the resulting image after a left-handed (indicated at the top of the Figure) off-axis spiral phase filter is displayed at the SLM. At the left side of the original bright field image a spiral phase filtered image of the SPP sample appears, in which phase- and amplitude edges are highlighted [1]. Note that due to the edge enhancement effect, the boundary of the field-of-view limiting aperture appears with a much higher contrast than in the corresponding zeroth diffraction order. Figure 3(g) shows the same sample section filtered with an off-axis right-handed spiral phase filter.

Note that a phase jump along a radial line of the SPP is visible in the experiment (Figs. 3(f) and 3(g)), which is not expected from the numerical simulation shown above. The reason for this behavior is that the used sample vortex plate is produced for 700 nm illumination wavelength, and not for the actually used 633 nm. Therefore the phase jump in the vortex plate differs from 2π, and exhibits the typical edge amplification effect of SPC microscopy for (non-integer) phase jumps.

In order to allow an optimal comparison, in Figs. 3(f) and 3(g) the SLM was used to display blazed holograms, diffracting only into the plus first diffraction order at the left side of the zero order image. Compared to the previously described method of displaying a binary hologram, this has the advantage that the two filtered images appear with optimal intensity at the same position in the camera plane, however, with the disadvantage that two successive images have to be recorded in order to allow a comparison between left- and right-handed spiral phase filtered images. Comparing Figs. 3(f) and 3(g) one already observes that the central singularity in the image of the sample SPP is suppressed in the first case, and highlighted in the second case. This difference becomes obvious in Fig. 3(h), which displays the resulting image after subtracting image 3(f) from 3(g). Consequently isotropically enhanced edges cancel out, whereas the central singularity is now distinctly highlighted. This identifies the imaged test SPP sample as a left-handed spiral phase filter, since it is converted to a Gaussian focused spot after filtering with the right-handed spiral phase filter.

The results of the simulations and the experiments clearly show the highlighting of the central singularity in the SPP test sample. This indicates that the method can produce a map of the chiral singularities in a sample, identifying the handedness of the singularity by the sign in the difference image.

3. Experimental results

As an example for a possible application we imaged the singularity distribution in samples of muscovite (mica fragment), and of acetylsalicylic acid crystals. The transparent samples with a thickness on the order of 10 µm were placed on the microscope stage and illuminated with the HeNe-Laser.

Diffraction of the image wave at the binary spiral phase hologram displayed at the SLM produces three separate sections within each image, corresponding to the first, the zeroth and the minus first diffraction orders, respectively, which are recorded in one camera exposure. The left- and right sections (a) and (c) correspond to left- and right-handed spiral phase filtered images (handedness indicated above) of the samples, whereas the central section (b) corresponds to the unfiltered image recorded in the zeroth diffraction order. The areas marked with a rectangle show the regions in which we investigate the phase singularity distributions by subtracting the left- and right-handed filtered images. The results are displayed in (d). The colors blue and red are used to highlight complementary oriented phase singularities.

The first two pictures of Fig. 4(1, 2) show samples of Muscovite consisting of only a few layers stacked on each other. Phase singularities only arise where adjacent screw dislocations are bunched together, forming a local “SPP-topography”. In the sample shown in Fig. 4(1), for instance, some screw-dislocations coalesce into a single spot forming a “macro-singularity”. It is clearly visible that in the left-handed spiral phase filtered image the singularity is suppressed, whereas highlighted in the right-handed spiral phase filtered image. As expected, after subtracting these two complementary filtered images from each other, the singularity becomes visible as a red spot in Fig. 4(d), depicting that the orientation of this screw dislocation is left-handed. In Figs. 4(2d) and Fig. 4(3d) more Muscovite layers are stacked on top of each other, which results in several singularities with different orientations, blue again corresponding to right-handed and red to left-handed singularities. The final experimental result, Fig. 4(4d), shows acetylsalicylic acid crystals. The method is seen to highlight the dislocations near the break-off edge of the crystal, where layers with varying thicknesses border each other.

 

Fig. 4 Mapping of the singularity distributions in various types of samples: (1-3) Muscovite, (4) crystallized acetylsalicylic acid. All images are recorded in a single exposure. The three sections in each frame correspond to a left-handed (a), a zero-order (b), and a right-handed spiral phase filtered image of the sample, respectively. The insets (d) shows the numerically calculated difference images of (a) and (c) for the regions marked in the images. Blue and red colored areas correspond to negative and positive values in the difference image, respectively, and thus indicate the orientation of the chiral phase singularities.

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4. Conclusions and outlook

A robust and easy-to-implement method to identify phase singularities in biological and mineralogical samples was demonstrated. We carried out measurements with lysozyme, ascorbic acid, Hematite, Silicon carbide, acetylsalicylic acid and Muscovite, but have restricted the paper to exemplarily presenting the results of only the latter two. The difference image between spiral phase filtered images with opposite filter handedness provides a map of the chiral singularities in the sample, which may be used to identify (structures in the) samples or to investigate growth mechanisms.

Although our demonstration setup uses transmission mode microscopy for transparent samples, the method can be straightforwardly adapted to image the surface of reflective samples in reflection mode microscopy, which might have applications in mineralogy, and material inspection, e.g. for semiconductor wafers.

Finally, let us remark that spiral phase contrast filtering has also been advantageously implemented in electron beam imaging [21]. In this case the method might in future be adapted to visualize screw and edge dislocations at a molecular level.