Orbital angular momentum light in microscopy

Abstract

Light with a helical phase has had an impact on optical imaging, pushing the limits of resolution or sensitivity. Here, special emphasis will be given to classical light microscopy of phase samples and to Fourier filtering techniques with a helical phase profile, such as the spiral phase contrast technique in its many variants and areas of application.

This article is part of the themed issue ‘Optical orbital angular momentum’.

1. Introduction

The special properties of optical vortex beams, i.e. paraxial beams with a helical phase profile which carry orbital angular momentum (OAM) [1–3], can be used in an impressive number of different ways, as can be clearly seen when browsing this theme issue. In holographic optical micromanipulation [4], for instance, phase masks with a helical phase profile are used to create ‘doughnut’ beams in order to trap dielectric particles, either in the dark centre or in the bright ring, depending on whether the refractive index of the particle is smaller or larger than the refractive index of the environment [5–7]. In this context, the OAM can also be used to apply an optical torque [8].

Helical beams have found applications also in optical imaging. A well-know example is the first stimulated emission depletion (STED) microscope [9], where a ring of light belonging to a beam with the phase distribution of an ℓ=1 Laguerre–Gauß mode was used to suppress fluorescence from molecules surrounding the scanning spot, thus improving the resolution beyond the diffraction limit.

In the present contribution, the various applications of OAM light and the special properties of optical vortices [10,11] in the broad field of microscopy will be discussed, with special emphasis on classical light microscopy. It will be shown that the usefulness of helical phase beams in optical imaging is not limited to the doughnut-shaped illumination of a STED microscope, but is reflected in a variety of techniques to improve the optical image in terms of contrast rather than resolution, one such example being spiral phase contrast (SPC) [12].

2. Spiral phase contrast imaging

Untreated biological samples often show very poor optical contrast. Therefore, a variety of optical methods have been developed to render small variations of optical thickness, i.e. changes in local refractive index and/or thickness, visible in an unstained ‘phase object’. A prominent example is phase contrast, invented by Frits Zernike in the 1930s. The light coming from a thin phase object illuminated by a plane wave may be approximated as , with , i.e. the undiffracted and the lowest-order diffracted light are π/2 out of phase. Phase contrast modifies this relative phase between diffracted and undiffracted light, to turn the phase modulation imposed by the sample into an intensity modulation. To this end, a phase-shifting element (typically a disc or a ring) is placed into the Fourier plane, where it shifts the relative phase by another π/2 (backwards or forwards for ‘positive’ or ‘negative’ phase contrast).

Thus, in a wider sense, phase contrast may be seen as selective phase-shifting of spatial frequency components in the diffraction pattern, and thus as filtering in the Fourier plane: according to Fourier optics [13], a lens performs a Fourier transformation (apart from a global phase factor) by ‘pulling’ the far-field from infinity into its focal plane. In this plane, referred to as the Fourier plane, under plane-wave illumination (called ‘central phase contrast’) the zero-order component or DC part, corresponding to the undiffracted light, lies in the centre, and the low-frequency components around it are pertinent to light diffracted from the gentle contours of the structures in the sample and the high-frequency components in the periphery correspond to the fine details (i.e. the ‘finest diffraction gratings’ with the largest diffraction angle) in the sample. Placing a programmable phase filter into this plane provides one with a means to selectively manipulate specific spatial frequencies, by phase-shifting, attenuating or blocking them. Blocking of the zero-order Fourier component, for instance, results in dark-field microscopy, where strongly scattering structures such as edges appear bright on a dark background. Shifting the phase of the zero-order Fourier component by π/2 with respect to the remaining wave emulates phase contrast.

(a). Fourier filtering with a helical phase mask

Of course the approach is not limited to this specific case, but there is a wealth of possibilities. Applying other types of phase-shifting masks may change the appearance of the sample image dramatically. In particular, if one uses a helical phase mask in a Fourier plane, some kind of phase contrast technique is implemented, which gives rise to brightly enhanced contours in phase samples. Because of the use of a helical or ‘spiral’ phase element this approach has been termed ‘spiral phase contrast’ (SPC) [12].

The generic set-up of Fourier filtering is a so-called 4f-set-up with two lenses, as shown in figure 1 for the special case of a helical or ‘spiral’ Fourier filter: lens 1 carries out a Fourier transform ; in its back-focal plane a helical phase pattern is imprinted on the wavefront by means of a spatial light modulator (SLM) [14] or by a helical phase plate [15,16], before lens 2 ‘undoes’ the Fourier transform, creating the image in its focal plane. The entire procedure on the object function E(r,φ), e.g. the electrical field transmitted by a thin phase sample, can be expressed as

2.1

This transformation is equivalent to the convolution (denoted by the symbol ⊗) of the function E(r,φ) with the kernel [17]

2.2

in polar coordinates (r,φ). This may be understood as an isotropic generalization of the Hilbert transform from one to two dimensions [12,18], i.e. a type of Riesz transform: in one dimension, the (negative) Hilbert transform [18,19] is defined as the convolution with −1/πx. In Fourier space, this corresponds to multiplying with the Fourier filter i sign(x), with the signum function

2.3

In view of this, applying as a (multiplicative) Fourier filter, which is equivalent to multiplying by a function flipping phase by π across the centre, has the effect of a Hilbert transform isotropically applied to all radial directions around the centre. In this sense, SPC is a radial extension of Hilbert phase microscopy [20]. The two-dimensional Hilbert (or Riesz) transform assigns a value of zero to the origin r=0. This means that in an experimental implementation of this transform, the filter has to be absorptive at the centre, where the phase singularity is located. As we will see, the centre, or in practice a small region around it, plays a crucial role in the appearence of the Fourier-filtered image.

Figure 1.

Image formation in spiral phase contrast. (a) The ideal point response of a single point of the object is not a diffraction-limited spot, but a diffraction-limited ‘doughnut’ with an annular intensity cross-section and helical phase. (b) For extended objects, each point has to be convolved with the ‘doughnut’-like point-spread function, which leads to the edge enhancement effect. (Online version in colour.)

In image processing, it is well known that the two-dimensional Hilbert (or Riesz) transform gives rise to edge enhancement when applied to a digital image. It has, for instance, been used for numerical processing of differential interference contrast (DIC) images for three-dimensional visualization [19]. SPC is an optical implementation of the edge-enhancing transformation, replacing the numerical post-processing. Early attempts to implement a spiral phase filter in an optical set-up had already been made in the 1990s [21,22], and in [18] a proof-of-principle experiment with an SLM was carried out, taking an aperture as the object, before the effect was independently found and interpreted in the context of digital microscopy in [12].

(b). Isotropic edge enhancement

It is important to get an intuitive understanding of why placing a helical phase mask in the Fourier plane leads to edge enhancement in the image. Figure 1 demonstrates the point response of a planar object when imaged with a spiral phase plate (or phase pattern on an SLM) in the Fourier plane. The point response of an optical system is characterized by its point-spread function (PSF), ideally given by a diffraction-limited spot. The spiral phase pattern imprinted on the wavefront in the Fourier plane between the lenses of the 4f-set-up, however, constructs the annular intensity that is typical for a beam with the helical phase signature of a Laguerre–Gauß mode with |ℓ|=1 upon propagation, as has already been mentioned in the Introduction when discussing STED. For a helical filter in a 4f-set-up, the PSF, i.e. the Fourier transform of e^iφ , gives a constant factor

2.4

in polar coordinates (see [23] for a derivation). This means that the point response is a diffraction-limited ‘doughnut’ with annular intensity and helical phase.

Fourier optics tells us that for a laterally extended planar object the image is formed by a convolution of the object transmission function with the PSF as the integration kernel [13], as already formally introduced in equation (2.2). For a thin object under homogeneous and coherent illumination, the edge enhancement effect can be made plausible by the following argument: in the convolution procedure the complex object amplitude is weighted with the shifted PSF at each point of the sample and then integrated over the entire aperture,

2.5

with polar coordinates defined by in the second expression. From this, it is clear that, in ‘flat’ regions in the sample, where neighbouring points around (x′,y′) are equal in amplitude and phase—and thus the factor E_in can be pulled out before the integral—the entire integral vanishes due to the integration of the azimuthal factor over 2π. Consequently, such regions of the sample appear dark. But especially at the ‘edges’ of structures in the sample, where the object has strong gradients in amplitude and/or in phase retardation, this cancellation is not perfect and a signal is generated. Figure 2 depicts this graphically. Overall, the result is a redistribution of the light into regions of the object where gradients in amplitude or phase occur. In the assumed ideal case, the phase along the contour of a phase step sample as depicted in figure 2 is determined by the geometrical orientation of the edge of the sample. The image intensity is isotropically redistributed within the sample, which makes the method very light efficient in comparison with dark-field imaging implemented as a high-pass Fourier filter.

Figure 2.

Graphical representation of the convolution of an object with a ‘doughnut’ PSF: in unstructured regions the integration over the helical phase profile of the PSF (which is π out of phase between any two points across the centre) leads to cancellation of the signal. Only in regions where either the phase retardation or the absorption of neighbouring points differs significantly, leading to a relative phase shift or a change in amplitude (see the inset), can a signal be created.

SPC uses ℓ=±1, but higher order vortices may be used for special effects. Filtering with ℓ=±2, for instance, highlights areas with curved contours [24]. Filtering with even higher order vorticities corresponds to selecting modes of higher order angular momentum. Digital spiral imaging [25] uses all spectral moments to obtain maximal information from the images.

An example of SPC imaging with an SLM in the Fourier plane is given in figure 3. It shows images of a thin phase object in the form of a transparent diffractive optical element (DOE) with a step-like thickness structure. Figure 3a corresponds to a bright field, i.e. it represents the resulting image in the absence of any structured Fourier filter on the SLM. Figure 3c,d relates to SPC, the spiral-phase filtered image of the DOE and the phase mask on the SLM used to generate it. Note that the central part has been made ‘absorptive’ by a fine diffraction grating in the middle of the phase mask which sends lights reaching this area elsewhere (see §4 for more details on this). The lateral size of the steps is of the order of 5 μm. Figure 3b corresponds to a dark-field image which was included for comparison. It is the result of a high-pass filter emulated by the same diffraction grating as for SPC, but without the helical phase pattern in the periphery. All images were acquired with a red LED (λ=638 nm central wavelength), with a condenser NA_illum=0.05 and a 40x magnification objective lens with NA_obj=0.75. The enhancement in brightness of the steps is obvious from the images. The images were taken under illumination conditions with partial spatial coherence and thus coherence artefacts are clearly visibile in the bright-field image. These can be seen to be somewhat mitigated in the SPC image, due to the very effective redistribution of the light. For more examples, including biological samples such as living cells, we refer to the literature (see [24] and references therein).

Figure 3.

Examples of Fourier-filtered imaging of the terraced structure of a transparent diffractive optical element by means of a red LED and an SLM in the Fourier plane.

(c). Pseudo-relief images

In the previous examples, the edge amplification of the spiral phase filter with an absorptive centre was demonstrated to be isotropic. This is to be expected, because the spiral phase plate—although it is not rotationally invariant—does not single out any specific direction or azimuthal phase angle. However, this symmetry can be broken intentionally, if one replaces the singularity in the centre of the filter by a small circular disc of homogeneous phase retardation. In a given set-up, the diameter of this central area has to be matched to the size of the central focused spot of the zero Fourier component. (In the off-axis version explained below, the central part is chosen to be a blazed grating without the ‘pitchfork’ singularity.)

In this arrangement, the focused spot of the zero-order Fourier component of the image wave is not removed any more, but develops into a plane wave, which can now act as a reference wave interfering with the image-carrying wave in the camera plane. Tuning the relative phase between the periphery and the centre of the spiral filter, either by changing the central disc or by rotating the phase plate, influences the characteristic shadow effects [26]. The apparent direction of illumination gives the pictures a pseudo-plasticity similar to images recorded with DIC microscopy. Examples of this effect are shown in figure 4. This approach obviously requires a certain degree of spatial coherence, but relief-like effects with a spiral phase plate have also been demonstrated for incoherent illumination by an LED in a Köhler illumination scheme [27], in order to reduce unwanted coherence artefacts, similar to those discussed before.

Figure 4.

(a) Spiral phase contrast filtering with relief effect: if the singularity in the centre is replaced by a circular region of constant phase value, the plane-wave zero-order light interfering with the filtered object field results in a shadow-like image. (b) The quantitative reconstruction of an epithelial cell from from three pseudo-relief images is shown. (Online version in colour.)

3. Spiral phase metrology

Changing the effective contrast is not the only possibility to use OAM light in microscopy. Helical beams may also be used to obtain quantitative information on the sample, in terms of local optical thickness profiles or a complex refractive index. As we will see, the ‘handedness’ of OAM beams can be used to our advantage. Moreover, helical beams may also be used to detect and classify vorticities in a sample, e.g. screw dislocations on the surface of a mineral. These applications will be briefly discussed in the following.

(a). Phase retrieval with vortex beams

A single-intensity image of a transparent object does not have enough information to infer the map of local optical thickness. In-line interferometry [28] provides a means to determine the quantitative thickness, but requires an interferometric set-up. Alternatively, methods have been designed to retrieve the desired information from several intensity images taken for slightly changed parameters, as in, for example, [29]. Another possibility is to use a type of ‘self-referenced’ interferometry, i.e. placing an SLM in the Fourier plane of a 4f-set-up and using it for phase stepping to record a series of images for the numerical reconstruction of the optical thickness [30].

A similar thing can be done with the pseudo-relief SPC imaging method mentioned earlier: one can change the apparent ‘illumination angle’ for the pseudo-relief SPC images (as in figure 4) by stepping the phase value in the central part of the hologram [26,31]. Choosing, for example, pseudo-illumination angles of φ₀=0,2π/3,4π/3 provides sufficient information to determine the phase profile by post-processing of these images (and also is a favourable choice, because the corresponding three unit vectors together cancel out the background). The exact algorithm is explained in [31]. Figure 4 shows the quantitative reconstruction of a human cheek cell. One can clearly infer that a fragment of a second cell is partly covering it, while in the individual non-quantitative images (not shown) this is hardly visible.

(b). Spiral interferometry

So far we have mostly assumed thin samples. What do we expect to happen when samples of an optical thickness of the order of a wavelength or more, such as an oil droplet, are imaged in the self-referenced interference set-up of the previous section (see figure 4)? It turns out that for (sufficiently smoothly varying) thick objects the contrast-enhancing shadow effects of thin objects evolve into a single, continuously spiralled interference fringe [23]. It can be shown (see appendix in [23]) that in this configuration the shape of the fringes depends not only on the phase distribution in the sample but also on the direction of the local phase gradient, characterized by a polar angle δ_ph. This effect can be explained as follows: for a smooth sample light is locally diffracted into the direction of the phase gradient. After focusing by the first lens (figure 1), the diffracted light is focused on the spiral phase plate (or SLM) at a position with polar angle δ_ph, thus acquiring an additional offset phase of δ_ph.

A closed isoline of equal phase around a local extremum, which would give rise to a closed fringe in standard interferometry, leads to a spiral-shaped fringe in spiral interferometry, because—going around the extremum—the local phase gradients along this isoline continuously acquire a phase from 0 to 2π (figure 5). The spiral convolution kernel breaks the symmetry and consequently the rotational sense of the spiral fringe allows one to distinguish between local maxima and minima.

Figure 5.

Spiral interferometry: for a thick phase object, such as an oil droplet, spiral phase filtering (with a central region for self-referenced interference as in figure 4) gives rise to spiral fringes. (Online version in colour.)

Spiral interferometry may be used to quantify the optical thickness of the sample in a more direct way than discussed previously: the spiral fringes may be interpreted as locations where Φ_n+δ_ph is constant modulo 2π. As the polar angle δ_ph of the phase gradient is always orthogonal to the local tangent along the fringe, one may also use the local tangent to the fringe, Φ_n=−α_tang up to multiples of 2π. Converted to length units, following the spiral fringe once around corresponds to an optical thickness of h=λ/Δn with Δn being the local difference in refractive index in the sample. For known refractive index, the topography of the specimen can be measured, or vice versa an unknown refractive index modulation can be determined, with excellent accuracy [32].

This provides a possibility to reconstruct the optical thickness map of a (smooth) phase sample from a single image [23] without the sign ambuiguity of conventional interferometry. The reason for the feasibility of this of course lies in the handedness of the OAM light, which breaks the symmetry and leads to a spiral fringe—spiralling clockwise or anti-clockwise depending on the local phase topography of the region [33]. This makes it possible to distinguish a local ‘hump’ from a ‘trough’ at one glance (after a calibration step to determine which spiral sense belongs to what in a given set-up with various optical components, which may affect the sign of the helicity). Figure 6 gives such an example, a polymer tape where material was displaced by local heating. The resulting bipolar landscape gives rise to a pair of spiral fringes of opposite handedness.

Figure 6.

Single-image demodulation from spiral fringes: (a) the spiral fringes are shown as recorded by the camera. (b) In spiral interferometry, the tangent to the central line of the fringes is proportional to the thickness h of the phase sample. This information may be used to create a space curve pulled out from the two-dimensional baseline to a distance depending on the orientation of the tangent (while keeping track of the number of turns corresponding to 2π in phase or to one wavelength) (c), which may finally be interpolated and smoothed to arrive at the reconstructed three-dimensional phase profile of the object (d). (Online version in colour.)

(c). Vortex mapping

Another interesting question is what will happen if the sample itself contains structures that give rise to optical vortices, for example a surface with screw dislocations, which are present in some minerals (e.g. mica) or many crystallized organic substances. Here, the special properties of OAM light give one a means to ‘chart’ the vortices in the sample and to determine their individual topological charge ℓ [34]. If the topological charge of the illumination light happens to be equal and opposite to the one generated by the sample, i.e. if ℓ_illum=−ℓ_sample, then they annihilate and the light focuses to a spot in the back-focal plane of an objective lens used for imaging. By using a ‘fork’ grating with ℓ prongs [35], two image-carrying beams travelling in different directions with opposite helicity are created. Here, one uses a binary grating in order to have the same intensity in the two diffraction orders with opposite helical charge. With the right choice of settings, the corresponding two images can be recorded simultaneously in two separate regions of the camera chip. This enables mapping of dislocations in the sample in a single camera exposure, as was demonstrated for various transparent samples [34].

Figure 7a shows an example using a binary pitch-fork grating for vortex detection. The binary grating gives rise to two image waves (the +1 and −1 diffraction order), which travel in different directions and are Fourier filtered with opposite helicity. Thus, the phase vortex of a spiral wave plate of topological charge ℓ=+1 can be detected as a localized spot in the diffraction order where it is annihilated by its counterpart created by the phase mask on the SLM. The other diffraction order contains a beam where the helical charge is increased to 2. Generalization to higher ℓ is straightforward. Another, very convenient, way of creating ‘vortex maps’ is to take two SPC images created by Fourier filtering with opposite topological charge ±ℓ and to calculate the difference of the two images (see figure 7b). Probing with increasing ℓ and incoherently adding the difference images populates the map of the sample with vortices of increasing order.

Figure 7.

Vortex mapping. It is possible to chart and quantify vortices in a sample by helical beam techniques, by using a binary fork-like phase mask on an SLM adding opposite helicities in each of the two diffraction orders (a) or by subtracting two spiral phase images of the sample made with opposite helicity from each other (b).

4. Experimental considerations

The actual experimental realization of the described methods deviates from the idealistic picture we have so far adopted for the sake of conceptual simplicity. A typical set-up for Fourier filtering microscopy will be more like the configuration shown in figure 8. There are several important issues concerning illumination, Fourier masks and scalings which will be discussed briefly.

Figure 8.

Experimental implementation of Fourier filtering with a phase vortex filter in the Fourier plane (off-axis configuration). (Online version in colour.)

In order to profit from the flexibility of a programmable device, the Fourier filter is often not a physical phase mask, but an SLM [14]. The term SLM is used for several types of systems for wavefront shaping, such as adaptive deformable mirrors and digital micromirror devices, where actuators or membranes are moved by electric or magnetic fields, or for a liquid-crystal-based system. The latter are miniaturized liquid-crystal displays (LCDs), which can dynamically influence the phase of light going through or being reflected from the panel when illuminated with light of a certain polarization. The active area typically is of the order of 2 cm² with up to full high-definition display resolution. The micrometre-sized pixels are addressed individually by a voltage, to create a phase shift in the range of 2π or more. Parallel-aligned (PAL) liquid-crystal panels allow for pure phase-only modulation, i.e. without changing the polarization of the incoming linearly polarized light. The phase masks can typically be refreshed at a 60 Hz rate; only ferro-electric SLMs typically allow for switching at kHz rates, but only between two phase values.

A first important difference from the idealistic set-up arises from the fact that SLMs are typically not transmissive but reflective, which leads to back-folded optical set-ups. Moreover, the Fourier filtering is mostly not carried out ‘on axis’, as depicted in figure 8, but one modifies the set-up to an ‘off-axis’ configuration [28,36], where—in order to get a cleaner image—one adds a blazed grating, i.e. a sawtooth phase structure where the phase rises linearly from 0 to 2π within one grating period, which has the property to (ideally) diffract an incoming plane wave into only one direction. When using an SLM, the blazed grating can be implemented by simply superposing the sawtooth phase pattern to the phase mask used for Fourier filtering modulo 2π. Its function is to send the image-carrying wave towards the camera, while the undiffracted background coming out at the angle of specular reflection, which contains no useful information on the sample, is blocked (figure 8). Especially for SLMs of limited diffraction efficiency, this improves the quality of the image, by increasing the contrast and suppressing the bright zero-order spot without discarding a substantial amount of light.

A disadvantage of the ‘off-axis configuration’ of Fourier filtering is that one has to somewhat restrict the field of view, a consequence of the necessity to separate the zeroth from the first diffraction order of the grating. On the other hand, it comes as an advantage of the off-axis configuration that it is possible to modulate both the phase and the amplitude of the SLM transmission function, by locally varying the modulation depth of the grating.

The convolution in equation (2.2) requires the centre (r=0) to be ‘absorptive’, which is not only a technicality but also has to be implemented. Physical phase plates can, in principle, be made absorptive, but SLMs here have a clear advantage: by leaving out the carrier grating in the central region of the phase mask, one can make this part ‘absorptive’ to a degree that can be chosen, e.g. for isotropic SPC or for dark-field imaging. Dark-field imaging is an extreme example in this respect, where the modulation depth of the grating in the centre is set to zero. (For on axis the situation has to be reversed, and one places a fine grating in the centre of the phase mask (figure 3)). Thus, to realize a spiral phase filter it is in principle sufficient to display the helical phase as a grey-level pattern on the SLM and centre it with respect to the zeroth-order Fourier components of the image wave, which is at the position of the focus of the illumination beam. The telescopic lenses in the set-up have the task of matching the beam diameter to the size of the SLM panel by controlling the magnification appropriately. Finally, the diameter of the central region has to be chosen to match the width of the zero-order diffraction peak of the light coming from and being diffracted by the sample (which is not to be confused with the zeroth-order diffraction of the blazed grating in the SLM plane). If this matching is done incorrectly, artefacts (such as halos) may arise.

The diffraction efficiency is an important factor for the quality of the images: for a digitally displayed blazed grating approximated by eight modulation steps the theoretical diffraction efficiency is above 95%, but the practically achievable diffraction efficiency is decreased by a variety of factors. Firstly, the discretization itself (both the pixel size and the number of phase step levels) has an important effect on the diffraction efficiency, but, secondly, also technical parameters (such as the fill factor) related to absorption and scattering losses play a role in state-of-the-art SLM devices. In total, in the off-axis approach the relative efficiency, i.e. the ratio of the shaped beam and the total light leaving the device, can be very good, of the order of 80%.

And, finally, the illumination also plays an important role. The ideal light for SPC is monochromatic and has a certain degree of spatial coherence, enough to allow for shadow and edge enhancement effects by Fourier filtering, but not too much to avoid coherence artefacts. Using a rotating diffuser plate illuminated by a HeNe laser has proven to be a good solution, because this combination allows one to tune to the desired optimal degree of spatial coherence by adjusting the beam diameter on the diffuser disc. Lately, high-power LEDs in combination with a narrow-band colour filter have become a practical, but not quite as flexible, alternative.

5. Spiral filtering in other contexts

The phenomenon of optical vortices is of course not limited to visible light. Thus, helical phase beams may also be used to create special effects in other types of imaging, such as electron microscopy or X-ray imaging. Isotropic edge enhancement has been investigated and successfully implemented also in soft X-ray scattering [37,38]. For soft X-rays of about 450 eV (or 3 nm wavelength), millimetre-sized binary (partly opaque) spiral zone plates made by lithography are used instead of (transparent) spiral phase plates. For matter waves, the principles are also largely applicable. Electron vortex beams have been created [39] and the principle of SPC has been demonstrated in electron microscopy [40,41]. In optics, combinations of SPC with other modalities are feasible and have been demonstrated: for instance, a combination with STED has been demonstrated, where the helical phase mask is additionally used for SPC without compromising the super-resolution of STED [42]. Even broadband illumination as used for low-coherence imaging or wide-field optical coherence tomography (OCT) can profit from the edge-enhancing effects of SPC [43], if the dispersive effects of the diffractive elements are compensated by an additional grating. Spiral Fourier filtering has also been applied in combination with photoacoustics, in order to enhance the contrast in Schlieren imaging of pressure gradients in photoacoustically induced shockwaves in a liquid [44]. Finally, ℓ=2 vortex beams have been applied for optical vortex coronography [45], where the phase singularity can effectively suppress the light of a bright source in the centre of the field of view (e.g. a star) and thus enhances the contrast for a dimmer object (e.g. a planet) at the periphery. Finally, as a last example, let us mention that the spiral filter effects also translate into the quantum world: it has been shown that, in ghost imaging with correlated photon pairs, a spiral filter in one arm leads to edge enhancement in the other arm [46]. And in digital spiral imaging, high-dimensional OAM entanglement can be used for non-local probing of phase objects [47].

6. Conclusion

Spiral phase contrast microscopy, i.e. Fourier filtering with a helical phase mask, has become an established tool for contrast enhancement, not just in classic optical phase microscopy but also in other imaging methods such as electron microscopy or X-ray imaging. If implemented by means of an SLM, one gains the extra benefit of high flexibility.

Figure 9 recapitulates various techniques explained in this paper in a simulation. Figure 9a depicts standard SPC, the object (which could be an amplitude or phase object) on the top, then the edge-enhanced intensity of the image and the phase weighted by the intensity. Note that the phase value along the contour is tied to its orientation. In figure 9b, the image at the top shows a phase object with a gradual increase in phase depth from left to right, which is mirrored in the strength of the SPC intensity picking up the phase step between the object and its surroundings (second image from the top). The third image shows the result of filtering with ℓ=2, demonstrating the bias to edges with non-zero curvature (and also featuring the striking sharpness of blocking out all light from the inner regions, e.g. from the panda’s ears, that is used in coronagraphy). Figure 9c gives three pseudo-relief images with a rotating shadow effect, as are typically used for quantitative SPC imaging, and figure 9d detects a phase vortex placed on the forehead of the (amplitude object of the) panda by calculating the difference between a left- and right-handed SPC image.

Figure 9.

The ‘panda panel’: simulations of spiral filtering methods for comparison at a glimpse (see the text for details).

With this we hope to have shown that OAM light has been put to good use in a variety of ways in optical microscopy. Probably, this will extend into the future in many unexpected ways.

Competing interests

The author declares that there are no competing interests.

Funding

Parts of this work were supported by ERC ADG catchIT (247024) and by the Austrian Science Fund FWF (P19582 N20).