WO2006072581A1 - Spiral phase contrast imaging in microscopy - Google Patents

Abstract

The present invention provides a microscopic device and a method for enhancing the edge contrast of an image observed with microscopic device. The microscopic device of the invention comprises a spiral phase element which is positioned in one of the focal or Fourier planes in the light path through the microscope thereby filtering the whole light field in said plane.

Description

Spiral Phase Contrast Imaging in Microscopy

The present invention relates to a microscopic device using filtering in the Fourier plane, in particular filtering with a spiral phase element for enhancing the edge contrast, for quantitative reconstruction of complex objects consisting of mixed amplitude and phase information, and for an advantageous interferometric measurement method.

In optical microscopy, many methods have been developed to increase the contrast of observed amplitude and phase objects. Known methods implemented in commercially available microscope systems include dark-field imaging, phase-contrast imaging, and differential interference contrast (DIC, also known as Nomarski method).

The dark-field and the phase-contrast methods are based on filtering of the image information in its Fourier plane. In dark-field microscopy, the zeroth order of the fully transformed image, i.e., the center of the two-dimensional Fourier transform (also denoted as the carrier wave), is blocked, allowing only light to pass, which is scattered by structures of the observed object. As a result, the object appears as a bright image in front of a dark background, leading to a significant improvement in the image contrast as compared to bright-field imaging.. In phase-contrast microscopy, the zeroth order is not blocked, but shifted by a quarter of a light wavelength with respect to the scattered light. Recombination of the zeroth order with the scattered light in the image plane then results in a transformation of a phase image into an intensity modulated image, i.e., each phase value of the observed object (measured modulo 2π) is substituted by a certain intensity level in the image. Further background information with respect to dark-field microscopy and phase-contrast microscopy can be found, for example, in G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, eds., Physical optical notebook: Tutorials in Fourier optics (SPIE Optical Engineering Press, Bellingham, Washington, U.S.A., 1989). The Nomarski or differential interference contrast (DIC) method is sensitive to phase gradients. The Nomarski method is a kind of shearing interferometry, based on the interferometric superposition of two slightly shifted images of the object. However, only edges with a certain local direction depending on the orientation of Nomarski prisms are amplified by the Nomarski method, i.e., the obtained Nomarski image is not isotropic. The Nomarski method employs two light waves with orthogonal polarizations. It is therefore possible that the image is distorted by a possible birefringence of the sample (see for example N. R. Arnison, K.G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microscopy 214(2004) 7-12).

J. A. Davis et al. report in Opt. Lett. 25(2000) 99-101 on image processing with the radial Hubert transform. In particular, they introduce a radially symmetric Hubert transform that permits two-dimensional edge enhancement. In G. A. Swartzlander, Jr., "Peering into darkness with a vortex spatial filter", Opt. Lett. 26 (2001) 497-499 and in K. Crabtree et al., "Optical processing with vortex-producing lenses", Appl. Opt. 43(2004) 1360-1367, further background information on Fourier plane filtering with so-called vortex lenses can be found. In these documents, macroscopic amplitude objects with 100% transmission contrast (macroscopic circular apertures and slits) have been investigated and edge enhancement has been reported.

hi US-A-5 262 889 a method and apparatus for frequency shifting an optical beam by the use of a phase plate or a spiral wheel is disclosed.

It is an object of the present invention to provide a new and improved microscopic device and a method for edge contrast enhancement in light microscopy. The device and method should further provide improved brightness of the observed microscopic image, hi particular, it is an object of the present invention to provide a device and a method which is sensitive to phase gradients independent of the direction of the gradients.

These objects are achieved with the features of the claims. In the microscopic device according to the present invention, the whole light field in a Fourier plane, i.e., a focal plane of the light path through the microscopic device, is filtered by a spiral phase element. That is, the whole light field and not only the zeroth order, i.e., the center of the Fourier image, are phase-shifted in the form of a spiral-shaped phase profile, preferably having the form exp(iφ), where φ is the polar angle in a plane transverse to the light propagation direction measured from the center of the Fourier image. Such a spatially varying helical phase offset can be obtained by diffraction of the Fourier image at a specially designed blazed phase hologram, which may be displayed at a high resolution spatial light modulator or by diffraction at a static phase hologram, or by using a spiral phase plate. The spiral phase element may be transparent or reflective depending on the position in the light path through the microscopic device. The spiral phase element preferably is centered with respect to the zeroth orders of the Fourier-transformed image wave by optical means. In principle, the spiral phase element can be positioned at any location where the illumination light (without a sample in the beam path) focuses.

The position where the spiral phase element can be positioned in can be shifted, for example, by using a light source which does not emit a plane wave but a convergent or divergent wave. It is even possible that the position of the spiral phase element is the objective of the microscope. In this case, it may also be possible to use a so-called "vortex lens". The position of the spiral phase element may also be moved to the backside of the objective of the microscope by varying the divergence of the illumination light. In this case, a transmissive spiral phase element would be used. Moreover, by varying the divergence of the illumination light, the position may also be moved far behind the objective of the microscope. This is similar to what is shown in Fig. 1 but without the need of the additional lens system 31 shown in Fig. 1.

The microscopic device of the present invention further comprises stage means for receiving a sample and a light source for illuminating the sample. The light source may be a laser diode or another source with a sufficiently large spatial coherence, such that the light can be focused to a sufficiently small spot in a Fourier plane of the beam path. The microscopic device of the invention further comprises optical means for imaging the light transmitted through the sample in case the sample is transparent or reflected by the sample. The optical means may comprise a microscope objective and may further comprise a lens system arranged such that the focal plane of the objective is imaged in the plane the spiral phase element is positioned in. The spiral phase element may have a singularity in its center or may have an area having a filtering property being a uniform phase shifting operation. In the latter case, the phase shift in the central area may be varied during detection of the image. With the method of the invention using a spiral phase element that has an area in its center providing a uniform phase shift, an oriented edge enhancement effect can be achieved generating a relief-like impression of an object which is obliquely illuminated by a spot light from a certain direction. A continuous rotation of the shadow effect creating a three- dimensional impression visualizing raises and depressions in the topography of a phase sample in an intuitive way can be generated, for example, by continuously shifting the phase of the central area and thus the direction of the shadow at video rate. Alternatively, such a phase shift can be also produced by just rotating an on-axis spiral phase element around its center by the desired phase shifting angle. Recording at least three images with different shadow orientations provides a means of numerical reconstruction of both, the quantitative amplitude- and phase information of the sample, using a subsequent numerical image processing step.

The same modified spiral phase method applied to a sample with a strongly pronounced phase profile produces a new kind of spiral-shaped interferometric map of the sample topography, which enables to reconstruct the sample profile from a single interferogram.

Interference of the zeroth order Fourier component, i.e., the central Fourier spot diffracted with a selectable phase offset at the center of the spiral phase element, with the remaining wave leads to interference fringes in the image in the form of a spiral. Following the course of such a spiral, after each 2π revolution, a new level which is one optical wavelength higher or lower, depending on the orientation of the revolution, than the starting point is reached. Thus, the topological indistiguishability between elevations and depressions at a smooth phase surface is broken, i.e., even a single interferometric image reveals the complete topology of the phase landscape.

The method of the present invention also allows to reconstruct the complete image information of a complex phase object, consisting of mixed amplitude and phase contributions in a quantitative correct way. Furthermore, the device provides a new method in interferometry, producing spiral shaped interferograms of a sample. These interferograms allow unique phase reconstruction out of one single interferogram, whereas in traditional interferometry there are at least three interferograms necessary. If higher resolution is desired, a phase-stepping method as in traditional interferometry can be applied, however, with the advantage of easier phase shifting in a self-referenced setup.

Preferred embodiments of the invention are described in the following in more detail with reference to the Figures.

Fig. 1 schematically shows a microscopic device according to an embodiment of the present invention;

Fig. 2 shows (a) a reflective, (b) a transmissive spiral-phase plate and (c, d, e) a spatial light modulator displaying different holograms to be used as a spiral phase element according to the present invention;

Fig. 3 schematically shows a device according to another embodiment of the present invention using white light;

Fig. 4 shows (a) a bright field, (b) a spiral phase contrast image obtained according to the invention and (c) a dark-field image of an absorptive sample imaged in transmission geometry;

Fig. 5 shows (A), (C) bright field images and (B), (D) corresponding spiral phase filter images of the same sample areas of a phase object;

Fig. 6 shows a simulation of the imaging of a circular phase step for the phase contrast method and for the spiral phase contrast method of the present invention;

Fig. 7 illustrates the dynamic shadow effect of edge enhanced phase structures of a sample object according to the present invention; Fig. 8 shows in (A) a simulated sample profile, displayed as a relief, in (B) a spiral phase contrast image of profile (A) and (C) an image of profile (A) obtained with a modified spiral phase contrast method;

Fig. 9 shows in (A) a simulated sample profile of a surface or a refractive structure, in (B) an interferogram of profile (A) obtained with a known method, in (C) a spiral interferogram of profile (A) obtained using the method according to the present invention, in (D) a single contour line of the spiral interferogram shown in (C), in (E) a processed contour line of the spiral interferogram shown in (C) and in (F) a surface profile reconstructed from the processed contour line shown in (E);

Fig. 10 shows in (A) an experimentally obtained interferogram using a method of the present invention, in (B) a spiral interferogram of the region shown in (A), in (C) a section of the spiral interferogram shown in (B), in (D) a single contour line of the spiral interferogram shown in (C) and in (E) a surface profile reconstructed from the contour line shown in (D);

Fig. 11 shows three methods according to the present invention for obtaining the phase shift necessary for a rotating shadow effect or for rotating interference spirals; and

Fig. 12 illustrates a method according to the present invention for evaluating the complete complex amplitude and phase information of a sample.

Spatial filtering of an image-carrying light wave with an amplitude or phase filter located in a Fourier plane of the image wave corresponds to a convolution of the image field amplitude with a kernel which is the Fourier transform of the filter. According to the invention, a phase filter in the Fourier plane is used, which imprints a "spiral staircase" on the phase of the incoming light wave, preferably of the form exp(iφ), where φ is the polar angle in a plane transverse to the light propagation direction, measured from the center of the Fourier image. Such a spiral filtering function has recently been proposed as a two-dimensional generalization of the one-dimensional Hubert transform (see K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform", J. Opt. Soc. Am. A, 18(2001) 1862- 1870). It has been shown that this kind of phase shifting in the Fourier space corresponds to a convolution of the image amplitude with a kernel function r^"2exp(iξ») in real space, where r and φ are the polar coordinates measured from the center of the filter kernel.

Convolution with such a kernel function replaces each image point in a point-symmetric environment or an uniform area with a zero. This is the result of the dependence of the kernel on the angular coordinate φ, which guarantees that any contribution to the convolution integral by one image point is cancelled by a negative contribution of a central-symmetric opposite point. The same reasoning suggests that the convolution produces intensity maxima at amplitude or phase edges of an image. Due to the spherical symmetry of the kernel, this applies for all directions of an edge within the object plane, i.e. the method is isotropic. The dependence on r^'2 concentrates the convolution process to the nearby surrounding of each point, such that the intensity enhancement at edges within an image falls off rapidly with increasing distance from the edge. Image filtering with such a kernel therefore results in an intensity distribution which is proportional to the intensity gradient of the original image, conserving the total image intensity. This promises extremely sensitive detection of phase jumps (or edges) within a phase object, which are orders of magnitude smaller than those detectable with the phase-contrast method.

Fig. 1 schematically shows a microscopic device according to the present invention. The microscope, which may be an inverted microscope 10, e.g. a Zeiss Axiovert 125, comprises a light source 1, stage means 2, an objective 31 and a mirror 32. In the object plane of the microscope 10, a transparent amplitude or phase sample is illuminated with an expanded light wave emitted from the light source 1, e.g. a single-mode 780 nm laser diode. In the Fig. 1, the propagation of the illuminating laser light is indicated. The transmitted light is collected by a microscope objective having e.g. a magnification of 2Ox or 4Ox. Behind the objective a further set 33 of two relay lenses is arranged such that the rear focal plane of the microscope objective 31 is imaged in a plane outside of the microscope. This plane is therefore conjugate with respect to the image plane, i.e. it contains a Fourier transform of the image amplitude information. There, the spiral phase element 5 is positioned. The image is observed using a detector, which may e.g. be a camera 4 comprising a lens and a CCD, the lens being located symmetrically (at a focal distance) between the spiral phase element and the CCD. Fig. 2 shows examples of spiral phase elements which may be used in the present invention. Figs. 2(a) and 2(b) show a reflective and a transmissive spiral phase plate. Fig. 2(c) shows a (transmissive or reflective) spatial light modulator (SLM) which displays a blazed phase hologram having a singularity in its center whereas Figs. 2(d) and 2(e) show the same hologram as Fig. 2(c), but with the central area replaced by a blazed grating having the filtering property being a uniform phase shifting operation. Figs. 2(d) and 2(e) differ by a different phase of the central grating, which controls the phase of the light diffraction from this region. In Fig. 1, a reflective spatial light modulator (SLM) is used, e.g. a Holoeye 3000 system, resolution 1920x1200 square pixels with an edge size of 10 microns, which can display blazed phase holograms which perform the selected filtering tasks. The SLM 5 is driven by a second graphics output of a computer, and it just displays copy of the computer monitor screen. Therefore holograms can be displayed in "normal" image windows and dragged, e.g. under mouse-control, across the screen. This allows very accurate centering of the holograms in the Fourier plane with respect to the incoming zeroth order Fourier component of the image wave, which is necessary for the spiral-phase method of the invention, but also for the dark-field and the phase-contrast methods, just by dragging the hologram window with a single-pixel resolution, e.g. 10 micron resolution, across the SLM screen.

The holograms are calculated as off-axis holograms such that the light diffracted into the first order can be separated from the other orders and is used for imaging. This results in an image which is "perfectly" filtered by the desired kernel functions, independent of the diffraction efficiency. For spiral phase filtering, a hologram is displayed which has a vortex discontinuity in its center (see Fig. 2(c)), which coincides with the focused zeroth order Fourier spot of the image light wave. The incident light wave is then diffracted from the blazed phase hologram displayed at the SLM with an absolute diffraction efficiency of approximately 40 %, and only the light diffracted at the first diffraction order is used for further imaging, after blocking undesired other diffraction orders (not indicated in the Figure). The image is then observed using a camera 5 comprising a third lens located symmetrically (at a focal distance) between the SLM and a CCD chip. It is noted that the system shown in Fig. 1 can be used straightforwardly to simulate any other phase and amplitude Fourier Filters, as e.g. a dark-field filter, by using an overall blazed grating which in a small circular area in its center is replaced with a non-diffracting unstructured area, thus discarding the center of the Fourier transform, or a phase contrast filter, by replacing the central area of the blazed grating with a π/2-phase shifted grating. The whole imaging setup can also be adapted to image the surface of nontransparent objects by impinging light from the laser diode at the object surface and imaging the scattered light with the microscope objective. Furthermore, it is possible to use transmissive spiral phase elements by modifying the arrangement of the spiral phase element and the detector.

Spiral phase contrast imaging using white light or broadband light from a pulsed laser source in an off-axis setup where spatial filtering is performed by diffraction at a grating, normally results in undesired dispersion effects. This dispersion, however, can be compensated by using a second reverse diffraction process at a grating with same period. An exemplary arrangement using this method is shown in Fig. 3. The setup shown in Fig. 3 comprises, in accordance with the setup shown in Fig. 1, a light source 1' which is a source providing coherent white light, stage means 2, an objective 31, a mirror 32 and a set of relay lenses 33. The hologram displayed on the SLM 5 is, however, split into two adjacent parts. Normal spiral phase filtering is performed in a first step at the upper part 5a which provides for a filter function with superimposed grating. The diffracted wave front is Fourier-back transformed by a further lens 51. A mirror 52 placed in the real image plane reflects the wave through the same Fourier transforming lens 51 to the lower part 5b of the SLM 5, where a plane grating is displayed, performing another diffraction process which compensates the dispersion introduced by the first process. Imaging is then performed as described with reference to Fig. 1 above, using a further Fourier transformmg lens and a camera 4.

As an exemplary sample object, a standard transmissive resolution target was used. Figure 4(a) displays a bright-field image of a letter within the sample having a size of 250 microns, which was recorded by just displaying a blazed grating as a phase hologram at the SLM in the setup shown in Fig. 1. Fig 4(b) shows an edge contrast enhanced version of the letter obtained according to the present invention by switching the SLM display to present a spiral phase hologram (e.g. the one shown in Fig. 2(c)), keeping all other settings of the imaging setup constant. A numerical comparison of the integrated intensities of the two images shows that they are equal, although strongly redistributed. For comparison, also a dark-field image of the sample was recorded (see Fig. 4(c)) by switching the SLM display to a blazed grating with a small non-diffracting sphere in its center, resulting in a suppression of light diffracted from the center of the hologram. The result resembles the spiral-phase filtered image of Fig. 4(b), however the brightness and the contrast are strongly decreased. This is a result from throwing away the zeroth Fourier component, which carries a large amount of the overall light intensity.

A major advantage of the spiral phase filter function becomes obvious by imaging phase sample objects with small phase jumps. An example is shown in Fig. 5. Two images of the same object section of a phase object, consisting of a scratch in a glass coverslip coated with water, are displayed. In (A) a transmission bright-field image was recorded by displaying a "normal" blazed grating at the SLM display in the Fourier plane of the setup, recording only the first order diffracted beam. In (B) the blazed grating was substituted by a hologram with a spiral-phase discontinuity in its center (like the one shown in Fig. 2(c)), and again the first order diffracted light was used for imaging. The two images were recorded with identical settings of illumination and image integration time. Obviously, the edge enhanced version in (B) obtained by spiral phase filtering shows much more details of the object plane section.

In (C) and (D) a bright-field image and a spiral phase filtered image of the same region of an oil-water boundary are displayed. Again the two images were taken under the same circumstances, i.e. the same illumination and camera settings. In this case, the boundary is barely visible with the bright-field method, but clearly recognizable in the spiral phase filtered image obtained according to the invention. It can be seen that the intensity of the edge of the oil-water boundary does not depend on its local direction, demonstrating that edge detection is isotropic. Quantitative analysis of the image pair reveals that the total light intensity, integrated over the whole imaged area is again almost identical for the bright-field and the spiral-phase methods. This means, that edge contrast enhancement of a spiral phase filtered image is the result of a redistribution of the image background intensity to phase edges of the imaged area. In order to compare the contrast enhancement features of the spiral phase and the normal phase contrast methods, numerical simulations of the two methods were performed. An example is shown in Fig. 6. The first image (A) shows the simulated outcome of standard phase contrast imaging of a circular phase step with a step height of 0.005π, corresponding to 0.25% of an optical wavelength, i.e. 1.5 nm in the case of 600 nm light. It turns out that the contrast, defined as the intensity amplitude (maximal minus minimal intensity) normalized by the maximal intensity is in this case 6%, i.e. the intensity modulation within the image is hardly recognizable. An analogous simulation using the spiral phase method of the invention was performed in (B). In this case, the contrast is 100%, i.e. the edge of the circular phase step is visible with full contrast even at the small modulation amplitude.

The simulation was performed by first Fourier transforming the sample phase image, then multiplying pixel by pixel with the respective filter function, and then performing the reverse Fourier transform. In the case of phase contrast imaging the filter mask consists of an array of "ones", with only one changed pixel in the center of the mask, consisting of the imaginary unit "i", corresponding to a relative phase shift of π/2 between the center and the other parts of the filter. The spiral-phase filtering mask consists of the spiral complex phase function exp(z^), where the central pixel (r=0) is set to 0. A calculation of the obtained image contrast as a function of the phase step height is shown in the right graph of Fig. 5. It turns out, that the contrast of the spiral phase method always equals one, i.e. the background is completely suppressed. The contrast of the phase contrast method oscillates, i.e. it first increases with increasing phase step height and it reaches full modulation at a phase step of π/2, but then it decreases to zero at a phase step height of π, since in this case the zeroth order Fourier component vanishes, hi the interval between π and 2π the behavior is repeated. The detailed behavior of the phase contrast method depends on the phase distribution within the chosen sample object, which determines the magnitude of the zeroth order Fourier component. On the other hand, the contrast of the spiral-phase method is not influenced by this effect. The numerical results suggest that, depending on the quality of the experimental setup, i.e. resolution and diffraction efficiency of the SLM, and centering of SLM with respect to the object Fourier plane, phase jumps on the order of 1% of the light wavelength or less should be detectable. As described above, image filtering in microscopy using a spiral phase hologram leads to a strong edge contrast enhancement of both, amplitude and phase objects. Similar to the established Nomarski method in microscopy, the effect is sensitive to the phase gradients within a sample, rather than to the absolute phase which is measured in phase contrast microscopy, or in interference microscopy. However, in contrast to the Nomarksi method the spiral phase method of the invention does not employ different polarizations, and it is isotropic, i.e. it highlights all phase edges of a sample object at the same time, independent of their local direction.

Spiral phase elements as those shown in Figs. 2(a) to (c) have a singularity in their center, i.e. the phase of the central point at r = 0 of such elements is not defined. Therefore, for the ideal case the singular point in the center has to absorb all incoming light at this position. Since the singular point in the center has no "area", this singularity is usually not important. However, this may not be the case if such a spiral phase plate is used as a spatial filter for Fourier plane filtering of an incoming, image carrying light wave. The reason is that the zeroth order Fourier component of the incoming light field (also denoted as the carrier wave) containing most of the intensity of the light field overlaps with the central area of the spiral phase plate. If the central point of the spiral phase element is really absorptive, then the overall circular symmetry is conserved and the spatial filtering operation results in isotropic edge amplification, as described above. However, if the spiral phase element is built such that the central point is a non-absorptive phase element, then its phase value has a significant effect on the appearance of the re-assembled light field in the image plane, similar to the importance of the phase of the zero-order Fourier component in phase contrast microscopy.

In the following, it will be shown that that such a spiral phase element with a non-absorptive center breaks the overall circular symmetry of the spiral phase operation, creating an oriented shadow effect with a direction determined by the phase of the central area of the spiral phase element, which can be useful in microscopy. The effect can be increased by expanding the singular point in the center of the spiral phase element to a sphere with a certain area whose filtering property consists in a uniform phase shifting operation of an incoming light wave. Qualitatively, the effect can be understood by examining the two-dimensional spiral phase filter in a polar coordinate system (with radial and angular coordinates of r and φ, respectively), and considering it as composed of an infinite number of radially proceeding one-dimensional Fourier filters at continuously changing polar angles, i.e. each single Fourier filter consists of a straight line passing through the center of the spiral phase plate. Each of these one-dimensional filters influences mainly the appearance of image structures proceeding orthogonally with respect to the corresponding filter "rays". Regarding one individual "filter ray" with the direction φ, the corresponding filter functions fix) consists of a signum function (defined as sign(x) = 1 for x > 1, siga(x) = -1 for x < 1, and sign(x) = 0 for x = 0), multiplied by a phase factor corresponding to the polar angle φ, i.e. f(x) = sign(x) exp(ϊφ), where x is the coordinate along a radial line with x — 0 denoting the center of the spiral phase plate. Using such a function as a (one-dimensional) spatial filter produces the so-called Hubert transform of the original function. In fact, the two-dimensional spiral phase transform has been recently introduced as the generalization of the one-dimensional Hubert transform.

The one-dimensional Hubert transform of a function looks similar to its derivative, i.e. it distinguishes between up- and down-stepping edges. A well-known example is the odd (with respect to its center) dispersion function obtained as the Hubert transform of an even Lorentz-function (i.e. an absorption line), and vice versa. However, the transform applies to the amplitude of the filtered function, whereas the recorded intensity distribution is even again. Nevertheless, it is possible to reveal the information about the sign of the field amplitude in the intensity distribution of the filtered function by superposing it mterferometrically with a plane wave. Fortunately, such a plane wave is generated "automatically" from the zero-order Fourier spot of the original function (image), if it is not absorbed in the center of the filter function. Note that the center x = 0 of the original Hubert transform kernel (sign(x)) is per definition zero, corresponding to complete absorption of the image carrier wave. However, in the experiment the spot at x = 0 is not absorbed, but just phase shifted by an adjustable phase offset θ. This single central spot in the Fourier plane evolves to a plane wave with a corresponding phase offset of θ in the image plane, which can interfere with the amplitude distribution generated by the "pure" Hubert transform. Depending on the interference term exp(i(φ-θ)) the interference can highlight positive or negative image amplitudes, or it can weight those amplitudes equally if the phase offset between plane wave and positive/negative amplitude parts is +π/2 and -π/2, respectively.

Returning to the two-dimensional spiral transform, the asymmetry of the shadow effect now becomes obvious, since the interference term exp(z((z>-#)) which determines the interference properties of up- and down-stepping edges in a direction perpendicular to the angle φ depends on φ itself, i.e. the same sample structure observed in different directions (e.g. by rotating the sample itself) appears differently, depending on its angle. For example, positive and negative contrast of a quasi one-dimensional up- and down-stepping edge is inverted if the sample is rotated by π around its center, and the difference between up- and down- stepping structures can be equalized if the sample is rotated by π/2. This is the same behavior one would expect from a sample illuminated by an oblique spotlight from a certain direction. Furthermore, this model explains that the direction of the apparent shadow can be controlled by adjusting the phase shift of the zero-order Fourier spot in the center of the spiral phase plate. The working principle for creating a dynamically rotating shadow effect without any mechanically moving parts is therefore to control the phase offset θ of the zero-order Fourier spot by electronically steering the corresponding pixel at the SLM phase filter.

Finally, the model explains why the direction of the shadow effect is rotated by 90°, depending on the property of a sample to be a phase- or an amplitude-contrast object. The reason is, that there is a well-known phase shift difference of π/2 between the zero-order and the higher order Fourier components of amplitude and phase contrast images. For example, this phase shift of π/2 is used in phase contrast microscopy, for generating an amplitude contrast image from a phase contrast object by just shifting the zero-order Fourier component by π/2. Concerning the shadow orientation which is given by the difference φ-θ, the actual phase of the zero-order component θ with respect to the higher order components is offset by π/2 between phase- and amplitude contrast objects, and this phase offset transforms into a corresponding 90° shift of the apparent shadow orientation. This phase difference between amplitude- and phase structures is the fundamental reason for the feature of the method of the invention to determine the complete quantitative complex image information of a sample by numerical processing of at least three images recorded with different shadow orientations. In the microscopic device according to the invention, the above-described effect can intentionally be amplified by increasing the central area (e.g. the number of SLM pixels) of the spiral phase plate. In this case the resulting Fourier filter resembles a combination of an ideal spiral phase element with a phase contrast Fourier filter, i.e. a phase shifter of the central area in the Fourier plane. That is, in the device shown in Fig. 1, the spiral phase element may e.g. be replaced by an SLM displaying holograms like those shown in Fig. 2(d) and 2(e).

Practically the holograms are calculated by first calculating the phase profile of a standard spiral phase plate, then attaching a certain phase value to a small circular area in the center of the phase mask, and finally superposing numerically an inclined plane phase term of the form Gxp(iG_xx+iG_yy) in order to produce a blazed off-axis hologram. If such a hologram is used as a Fourier filter, then the resulting edge enhanced filtered image shows a shadow effect, similar to the effect of oblique illumination from a certain angle. Particularly, for a certain direction, edges within the image with a positive phase slope have another intensity than edges with a negative phase slope of the same magnitude. In the transverse direction, the two (up- and down-stepping) edges have the same intensity. This difference in illumination between up- and down-stepping edges, which happens only in one certain direction is similar to the shadowing effect of a slanted spotlight illuminating the sample plane from one side. The hologram displayed at the SLM (and particularly the phase of the blazed grating displayed in the circular control area) can be changed dynamically at video rate, which can be used to create dynamic effects in real-time microscopy, as e.g. to excite the impression of a sample which is illuminated by a spotlight rotating around it.

It is noted, that spatial filtering can be also performed with non-holographic filters, like a spiral phase plate, or by on-axis diffractive optical elements. However, the off-axis holographic approach provides more flexibility and better performance for non-ideal phase modulators, since the off-axis diffracted light contains perfectly filtered information even for non-perfect modulation performance of a SLM (the SLM phase modulation properties only influence the diffraction efficiency, but not the wave-front of the diffracted light). Furthermore, an off-axis holographic phase modulator can perform both, spatial amplitude and phase filtering, since the diffracted phase is influenced by the local phase of the hologram fringes, whereas the diffracted amplitude can be controlled by spatially modulating the diffraction efficiency at different positions of the SLM by controlling the phase modulation depth of the SLM.

The directional dependence of the effect can be manipulated to appear in any desired direction by changing the phase of the grating in the central area within the hologram. In an experiment, a movie sequence of 16 images of a fingerprint on a glass coverslip as a sample object has been generated with stepwise changing phase of the central vortex area. The resulting movie can be play-backed in real-time (video-rate) at the SLM as a dynamically changing vortex filter. As a result, sample areas imaged with this method appear to be illuminated obliquely by a spotlight, rotating around the sample plane. Four out of these 16 images are shown in Fig. 7(a) to (d). This dynamically changing impression which can be produced in real-time without using any mechanically moving components gives a useful quasi-three-dimensional overview of the observed sample. Particularly, it is possible by the shadow effect to distinguish between rising and falling edges, e.g. to see at one glance whether a structure is a depression or an extension within the sample topography. Furthermore, it is also possible to distinguish between sample edges with amplitude or phase contrast, or a mixture of both. The reason is that the rotating shadow has a phase offset of 90° between objects with amplitude and phase contrast, i.e. if a phase edge within a sample appears to be illuminated from the left, an amplitude edge would simultaneously appear to be illuminated from above. The reason is based on the different diffraction properties of amplitude and phase objects, i.e. the phase offset between the zeroth order Fourier component, and the scattered Fourier components of amplitude and phase objects has also an offset of 90°, which translates into a phase shift between the corresponding shadow effects.

A numerical simulation of these effects is displayed in Fig. 8 which shows a simulated sample profile, displayed as a relief. The lower left structure is absorptive, the other structures are refractive. In (B), a "normal" spiral phase contrast image obtained with a spiral phase element having a central singularity resulting in isotropic edge amplification is shown. In (C), an image obtained with the modified spiral phase contrast method, with a transmissive center of the spiral phase element is shown. The shadow-effect produces a relief-like view, which distinguishes between elevations and depressions of the sample topography. It is noted that the apparent shadow direction of the absorptive structure (lower left corner) is rotated by 90 degrees with respect to the other refractive structures). A further advantage of the controllable shadow effect in edge contrast enhancement is the fact, that it provides the key to a huge increase in detection sensitivity by a subsequent digital image processing step. In the easiest case, a digital subtraction of two images recorded with opposite shadow phases results already in a strong edge enhancement, and background suppression. The reason is that the shadow effect has both, positive and negative intensity contributions to the background intensity of a recorded image, at raising and falling edges, respectively, i.e. one kind of edges appears brighter, and the opposite kind darker than the image background. A subtraction of two images recorded with opposite shadow phases then removes the unmodulated background, whereas the edges with their switching contrast are strongly amplified.

This digital image processing method can be generalized to process a whole sequence of N images recorded at equally spaced shadow phase values ψ_\ ... (p^ m^' . the range between 0 and 2π. The stored real intensity images are then transformed to complex images by multiplying each image with its corresponding phase factor, i.e. the k-th image with Qxp(iψ_k). Afterwards all complex images are summed and the absolute value of the result is displayed. This resulting image is a generalization of the two image subtraction method and looks like an isotropically edge enhanced version of the original sample, i.e. without shadow effect, similar to an .ideally, spiral filtered image with an absorbing pixel in the center of the spiral phase filter. However the contrast of the resulting image is even more increased, and background noise is significantly suppressed. Furthermore, the information about rising and falling edges, and the difference between images with amplitude and phase contrast is still conserved in the resulting complex image, and can be retrieved in a qualitatively exact way by numerical processing of the resulting image performing a numerical two-dimensional Hubert-transform, as indicated in Figure 12.

The method of producing dynamic shadow effects explained before has also an interesting application in interferometric imaging of phase with a phase topology exceeding the optical wavelength. Numerical simulations and experimental data show, that interference of the zeroth order Fourier component, i.e. the central Fourier spot diffracted with a selectable phase offset at the center of the vortex hologram, with the remaining wave leads to interference fringes in the image, with a periodicity corresponding to an optical wavelength. However, in contrast to "traditional" interferometry, the interference fringes are not closed lines which resemble to the contour lines in a map, but they consist of a single line in the form of a spiral. Following the course of such a spiral, one reaches after each 2π revolution a new level which is one optical wavelength higher or lower, depending on the orientation of the revolution, than the starting point. This has the advantage over "traditional" interferometry, that the topological indistinguishability between elevations and depressions at a smooth phase surface is broken, i.e. even a single interferometric image reveals the complete topology of a phase landscape. This has been experimentally checked that the development of the spiral is suppressed, if the zeroth Fourier component of the imaging light wave is blocked. In this case, an "ordinary" interference pattern consisting of closed contour lines evolves, like in normal interferometry. The exact orientational position of the spiral depends on the programmable phase between the zeroth order Fourier coefficient and the remaining wave front, which can be adjusted by the method described above. This provides the additional possibility of introducing a spiral rotation by Fourier filtering with a holographic movie identical to the one explained above. This is illustrated in Figs. 9 and 10.

Figure 9 shows in (A) a simulated sample profile of a surface or a refractive structure. In (B), a "normal" interferogram is shown. Closed contour lines do not distinguish between elevations and depressions. In (C), a spiral interferogram, obtained by filtering with the modified spiral phase contrast method, according to the invention is shown (with a transmissive center of the spiral phase element). Depending on the topography, the spirals change their rotational direction. A single contour line of the spiral interferogram is shown in (D), whereas (E) shows a processed contour line. The local direction of the line is proportional to the surface height, modulo one wavelength. This already allows to assign a unique height to each single point of the contour line. (F) shows the reconstructed surface profile which results from the contour line shown in (E) by fitting the surface at the sampling points given by the contour line.

Figure 10 (A) shows an experimentally obtained interferogram of oil drop smears at a glass coverslip. The image was filtered with an "ideal" spiral phase element, with an absorptive singular point in the center, resulting in "normal" contour-like interference fringes. (B) shows a spiral interferogram of the same sample region obtained after filtering with the modified spiral phase element (blazed grating in a small central area, instead of an absorptive spot). In (C), a section of the spiral interferogram which will be processed is shown which may result in a single contour line of the spiral interferogram as shown in (D). (E) shows the reconstructed surface profile obtained by processing the contour line shown in (D) and fitting the surface at the obtained sampling points.

The interferometric method of using only one spiral-interferogram for unique reconstruction of the object topography has advantages in high peed interferometry, e.g., with laser pulses, or video-interferometry. However, for achieving maximal precision phase-stepping methods are advantageous, using more than one interferogram recorded with shifted fringe phases. Such a phase-stepping can be performed with a method in analogy to the production of the rotating shadow effect, i.e., with one of the methods depicted in Fig. 11. Fig. 11 shows a method for obtaining a phase shift necessary for a rotating shadow effect or for rotating interference spirals. The grey- values within Fig. 11 correspond to phase values in the range between 0 and 2π in the setup. As shown in Fig. HA, the phase shift can be achieved in an off-axis setup by diffraction from the grating, where the phase of the grating in the central inner sphere is shifted. As shown in Fig. 1 IB, the phase shift is achieved in an on-axis setup by offsetting a phase of the center sphere from one image to the next. Finally, as shown in Fig. HC, the phase shift is achieved in an on-axis setup by rotating the whole spiral phase element from image to image by an angle which corresponds to the desired phase offset.

Post-processing of a sequence of at least three interferograms with different fringe phases provides a means for quantitative evaluation of the complete amplitude- and phase profile of the investigated sample. Fig. 12 illustrates how by recording at least three images with different shadow orientations, a means of numerical reconstruction of both, the quantative amplitude- and phase information of the sample can be provided, using a subsequent numerical image processing step. To this end, at least three shadow-effect images recorded with equidistant phase offsets can be numerically processed to obtain a resulting image with suppressed noise, which contains quantitatively correct the whole amplitude and phase information of a complex sample, i.e., a sample with a mixed amplitude and phase information. The processing comprises adding the at least three shadow-effect images and subjecting the sum to a reverse 2D-Hilbert transform. The same method applied to a sample with a deep phase profile, resulting in interference spirals within the individual images, provides a quantitatively correct means for an interferogram analysis of complex samples. The method may be generalized to process a series of more than three images which again are preferably recorded with equidistant orientations, resulting in an increasing accuracy of sample reconstruction.

In summary, it is possible with the method of the present invention to achieve edge enhancement which may be isotropic when a perfect spiral phase plate with a singularity in its center is used, or may be unisotropic when using a modified spiral phase element wherein a central area has a plain phase shifting property without spiral modulation as described above with respect to the shadow effect. With method of the present invention, spiral shaped interferograms from samples can be obtained which allow unique phase reconstruction from only one single interferogram. The present invention further provides a method for rotating the shadow direction or the orientation of the interference spirals, using for example the possibilities indicated in Fig. 10. The invention further provides a method wherein three or more of the images or interferograms are used, which are recorded with different shadow- directions or spiral fringe phases to evaluate both, the complete complex amplitude- and phase information of a sample in a quantatively exact way.

A detailed discussion of the analysis of spiral interferograms is given in the document enclosed as Annex which is to be published in the Journal of the Optical Society of America A. This Annex forms part of the disclosure of this specification.

Spiral interferogram analysis

Alexander Jesacher, Severin Fϋrhapter, Stefan Bernet, and Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University, Mullerstr. 44, Λ-6020

Innsbruck, Austria

Interference microscopy using spatial Fourier filtering with a vortex phase element leads to interference fringes that are spirals rather than closed rings. Depressions and elevations in the optical thickness of the sample can be distin- guished immediately by the sense of rotation of the spirals. This property al- lows an unambiguous reconstruction of the object's phase profile from only one single interferogram. The present paper investigates the theoretical background of "spiral interferometry" and suggests various demodulation techniques based on processing one single or multiple interferograms, respectively.

The properties of a spiral phase element as spatial filter in imaging applications has been discussed in several recent publications.^1-7 Placing the filter in a Fourier plane of the optical system leads to strong edge contrast enhancement for amplitude and phase objects, similar to the Nomarski or DIC technique.⁸ In a recently published article⁹ we considered the ap- plication of a phase vortex to interferometry. We reported the observation of spiral-shaped interference patterns that appear when thick phase samples are examined. A specialty of such spiral interference pattern is that they contain sufficient topographic information to allow the reconstruction of the object's phase profile. Conventional methods¹⁰ typically re- quire three different interference patterns in order to eliminate the ambiguity between a step "up" or "down" from a given contour line to reach its neighbor.

The present paper investigates the theoretical background of " spiral interferometry" and suggests methods for the demodulation of the coiled fringe patterns. On one hand, it is demonstrated how the overall topographic information of a sample phase profile can be eval- uated from a single interferogram. This might have applications in high speed interferometry with a single laser pulse, or in the feasibility to record an interferometric video movie of a rapidly varying surface, where each video frame can be post-processed separately. On the other hand - if imaging speed is not of primary importance - the setup allows high accuracy object reconstruction from a selectable number of interferograms, which can in our approach easily be produced using a straightforward non-mechanical phase-stepping method. 1. Vortex filtering

Fig.l shows the generic experimental setup, which is a so-called 4-f-system. The object of interest in the (x, j/)-plane is illuminated with sufficiently coherent light. A convex lens performs a Fourier transform of the field distribution E_in(x, y). After influencing the image by a vortex phase filter, a second Fourier transform is accomplished by another, identical lens. As indicated in Fig.l, the phase vortex is slightly modified, i.e. its center is replaced by a circular area of constant phase shift. The reason for this modification is our intention to use the zeroth order spot of the object light field as reference wave for producing so called "self-referenced" interferograms (see Ref. 9 for details of the set-up), which are captured in the plane (x', y'). Since the zeroth order light focusses in the center of the spiral filter, it is now phase shifted by a constant value (determined by the phase of the central circular area) and becomes a plane reference wave after the second lens, i.e. in the image plane. In our experiments we use the LC-R 3000 reflective liquid crystal spatial light modulator (SLM) for generating the filter functions. Its small pixel size of 10 microns allows us to match the diameter of the central disc area to the size of the zeroth order spot. The following mathematical considerations at first investigate only the effect of pure spiral phase filtering, i.e., they neglect the mentioned modification, assuming the Fourier filter to be a perfect phase spiral. The effect of the superposition with a phase controlled plane wave, given by the zeroth Fourier order, is then considered later.

According to the Fourier convolution theorem, the result of the vortex filter process can be derived by a convolution of the original object function E_in(x, y) with the Fourier transform of the filter:

One has to use e mirrored filter function exp (iθ), which is identical to — exp (iθ), due to the inverting property of the 4-f-system. The Fourier transform of — exp (iθ) is also called the convolution kernel Ky. In polar coordinates it has the form:

Ji is the first order Bessel function of the first kind, λ the light wavelength and / the focal length of the two lenses. The explicit analytical form of Ky (r, φ) is derived in the appendix. It is related to the field distribution of the Laguerre Gaussian Mode TEM_Q1 - which is also known as "optical vortex" or "doughnut mode" .¹¹ In the limit simplifies to

For comparison, Eq.3 describes the convolution kernel of the setup shown in Fig.l without a vortex filter - hence representing a simple two-lens imaging system:

K(r, φ) represents the point spread function (PSF) of a circular aperture with radius p_maχ- Comparing the convolution kernels of Eq.2 and Eq.3 (see also Fig.2), one can identify the main differences to be the different orders of the Bessel functions and the vor- tex phase factor exp (iø), which causes the vortex kernel Kγ to be ζό-dependent. This anisotropy is responsible for the spiral interference patterns, as will be shown in the following.

A descriptive way to achieve the convolution of two functions Ky {x, y) and Ei_n(x, y) at a certain point P = (P_x, P_y) is to mirror Ky at the origin, then shifting it to point P, and finally integrating over the product of the shifted kernel with E_in. Consider - for simplicity - an approximated filter kernel

where N is a scali factor defined as

This assumption allows the result of the convolution at point P to be derived as

Here (rp, φp) defines a polar coordinate system with its origin in the center of the kernel (see Fig.3), and Ei_n the input light field, expressed in this local coordinate system. For an investigation of the basic effects of such a convolution procedure, we expand in a Taylor series to first order:

Here g_Am

describe amplitude and phase gradient of Ei_n, evaluated at point P (= kernel-center), which is equal to the origin 0 in the local coordinate system i^χp, Vp)- Together with Eq.5, the output light field can - after integration (see appendix) - finally be written as:

Here, 5^_m(P) and 5p_ft(P) are the polar angles of the corresponding gradients (see Fig.3). Eq.7 allows a qualitative examination of the vortex filtering properties: It is apparent that it consists of two terms, which describe the effects of amplitude and phase variations of the input object on the filter result. The terms are proportional to the absolute values of the gradients g_Am and gp_h respectively, which explains the observed strong isotropic amplification of amplitude and phase edges. The factors exp (iδph) and exp (ό^_m) can be interpreted as the manifestation of gradient-dependent geometric phases in the following sense: δp_h and δ Α_m ^were originally geometric angles which indicated the spatial direction of the respective gradients. These factors now appear in the filtered image not as directions anymore, but as additional phase offsets of the image wave at the appropriate positions. Similar to other manifestations of geometric phases,¹³ the phase offset does not depend on the magnitude of the amplitude or phase gradient, but only on its geometric characteristics, which in this case is the direction of the field gradient.

A consequence of the factors exp (iδp_h) and exp (iδ_Am) is the fact that the edge amplification becomes anisotropic when interfering E_out with a plane reference wave, because they produce a phase difference of TΓ between a rising and falling edge of equal orientation. The resulting "shadow effect" gives the impression of a sidewise illumination, in accordance with our earlier observations.⁷ Due to the factor i in Eq.7, this pseudo-illumination shows a 90-degree rotation between pure amplitude and phase samples.

In the following discussion we assume the object to be a pure phase sample (with a thick- ness in the range of the illumination wavelength or larger). Thus we can neglect the first term in Eq.7. In this case, the anisotropy in edge enhancement which emerges when superposing the filtered waυefront with an external plane reference wave, enters the interference fringes: The shape of the fringes depends not only on the phase distribution φ_in of the sample, but also on the direction of the local phase gradient. In the vicinity of local extrema, the fringes are spiral-shaped, and the rotational direction of the spiral allows to distinguish between local maxima and minima.

In view of interferometry, the weighting of the resulting amplitude by gp_h is unwanted, because of its influence on the fringe positions. However, this effect is - according to our ex- perience - of little importance, except very close to local extrema and saddle points; moreover it can be completely eliminated by using multiple spiral images for object reconstruction (see chapter 2). To get a better understanding of how the vortex filter alters phase profiles, Fig.4 shows the result of a numerical simulation. It considers a Gaussian-shaped phase sample as object of interest (left image). As it is apparent in the figure, the vortex filtered wavefront (right image) looks somehow similar to the original, but with an "imprinted" phase spiral. 2. Demodulation of spiral interferograms

The analysis of conventional interference patterns is usually based on three single interfer- ograms - each showing the same object but taken at different values for the phase of the reference wave. The images therefore show slightly shifted fringe patterns, such that together they contain the whole topographic information of the examined sample. Mathematical com- bination of the three single images yields the object's surface structure modulo 2τr, which can finally be unwrapped using one of several phase unwrapping algorithms. Spiral interfer- ograms - in contrast - allow feasible object reconstruction based on only one single spiral filtered image. To the best of our knowledge this efficient phase surface reconstruction "at a glance" is a novelty in interferometry.

2.A. Single-image-demodulation

The presented demodulation techniques are based on the assumption that the filtered wave- front is of the for

with constant field amplitude. Gonse- quently, the fringe positions reflect the local values for ψi_n + δp_h, modulo 2τr, which implies that mod[ipi_n + δp_h, 2π] is constant at positions of maximum fringe intensity. The value of this constant can be selected in the experiment by adjusting the phase of the zeroth Fourier component (center of the spiral phase plate).⁹ Since δp_h is always perpendicular to the tangent of the local spiral fringe, characterized by the angle a_tan, we obtain the relation Setting the arbitrary constant C to zero, we may write

Scaled to length units, this yields

where Δn is the difference between the refractive indices of object and surrounding medium, and λ the illumination wavelength.

According to Eq.9, a basic idea to obtain the topographic information of a pure phase sample from the interferogram is to process the spiral from one end to the other - continuously assigning height information h to each point, which is determined by the angle of the spiral's tangents. In the following, two variants of single-image demodulations methods are presented, which differ by the way they represent interference fringes by single lines.

2.A.I. Contour line demodulation

A quite simple way to obtain the surface profile is based on processing contour lines. Fig.5 demonstrates the process considering a practical example. 5B shows the spiral interferogram of a deformation in a transparent glue strip. Phase modulations of such a shape emerge due to internal stresses, when the rigidity of the film is decreased by local heating-up. The opposite coiling directions of the spirals in the interferogram indicate, that the deformation consists of an elevation adjacent to a depression.

In a first step one single contour line (Fig.5c) has to be constructed, which is a closed line and connects points of equal intensity in Fig.5b. This can be done using standard image processing software. In our case, the software represents contour lines by an oriented array of L pairs of corresponding (x, ?/)-vectors (L being the length of the line), which makes their handling quite simple, because the height assignment according to Eq.9 can be carried out pointwise, following the contour line array from its beginning to its end.

Unfortunately, this process generates a systematic error, since after assigning the height information, i.e. after adding a third "height" -dimension, the resulting curve always shows a discontinuity: Starting and ending points of the numerical processing - which are direct neighbors in the two dimensional contour line - are now separated in height by a step of An λ. This is a consequence of the amount of clockwise and counterclockwise revolutions, which necessarily differ by 1 in a closed two dimensional curve. However, this error is tolerable in many cases, where an overview over the topography of an extended phase object is required with a minimum of computational expense. In addition, the error can be corrected by further processing, e.g., by cutting certain parts out of the line, as demonstrated in the example of Fig.5. In the following, an alternative way of single-image demodulation is presented, which avoids this systematic error.

2.A.2. Center line demodulation

In this demodulation method, the spirals are represented by curves, which follow the inten- sity maxima. They are constructed from the spiral interferogram by applying an algorithm, which continuously removes pixels from the spiral boundaries, until a "skeleton" remains. As a result, one obtains a connected line following the maxima of the spiral fringes, which is further processed similar to the contour line method described in the previous section, i.e., the local tangential direction is calculated and transformed into height information. In con- trast to the contour line method, the maximum intensity line is avoiding the artificial phase jump. However, the calculation of the maximum intensity positions cuts a line into two parts at possible branching points of the fringes. After calculation of the height characteristics for each branch according to Eq.9, they are finally reconnected. This procedure ensures a quite accurate reproduction of the profile (see Fig.6b). In a final step, interpolation can be used to construct a continuous surface (Fig.6c).

The center line method provides a more accurate reconstruction of the object phase profile from a single interferogram, compared to the contour line method described before. This advantage is, however, at the expense of more computational effort, since a contouring algo- rithm is usually available in software packages, a "fringe tracking" algorithm maybe not.

2.B. Multi-image- demodulation

If highest surface reconstruction accuracy is desired, a method based on processing numerous interferograms can be devised, which is based on processing more than one image, each showing an interferogram captured at a different value for the phase of the reference wave. The benefits include more accurate results and the applicability to objects of nonuniform transmission. As will be shown in the following, nearly exact results are obtainable by including three (or more) different interferograms.

The intensity of a general interference pattern can be described as

Here the reference wave parameters, A_ref and ^Ve_/, are assumed to be constants. In our case⁹ the reference wave is the zeroth order of the object itself, which ideally forms a plane wavefront of uniform intensity in the image plane (x', y'). The reference phase φ_ref is thereby adjustable by adding appropriate phase shifts to the center region of the filter hologram. In the case of an off-axis hologram the center region is replaced by a blazed grating of adjustable phase off-set.

After acquisition of three interferograms at different reference phase values ^, in a first step the reference phase information is added to the intensity patterns by multiplication with exp (iψj). In the event of the interferogram Iχ captured at the reference phase value φι, this yields a complex image of the form I_cχ — Iχ ^■ exp (iφi). The arithmetic mean /_c of the three complex images I_cj yields

provided that the reference hase values φj are evenly distributed within the interval [0, 2τr], that is

now includes the complete object topography (the factor A_re_f can also be determined from the three interferograms - see appendix).

However, strictly speaking this is only true for interferograms created by a separate external reference wave. In the case of self-referenced interferometry, we must keep in mind that the object's zeroth order itself represents the reference beam. According to Eq.10, this means that A_oy exp (iφobj) describes the object without its zeroth order. Consequently, Are_f has to be added for a complete image reconstruction. Practically, for achieving best results, it is necessary to alter the zeroth order amplitude until the resulting surface appears smoothest. The experimental reason for this discrepancy is that the reference wave is in practice not completely identical to the zeroth order spot of the object.

So far, the demodulation does not already result in the original sample, but in its vortex filtered image (see Fig.7). At this point, different ways of further numerical image processing are imaginable: The first one, which is explained in Fig.7, consists of an inverse spiral filter process, which means numerical spatial filtering with the function exp (~iφ), in order to get the true object profile, followed by image unwrapping. It compensates all errors caused by the spiral filter, i.e., the influence of object phase variations on the amplitude of the light field and vice versa (see Eq.7).

An alternative way could again make use of contour lines: One gray-level in the vortex fil- tered image can be chosen to achieve the height deconvolution analogue to the single-image techniques. The complete topography can finally be restored by either repeating this process for every gray level or by matching the remaining parts of the image to the contour line, whose height characteristics has formerly been acquired. Although this procedure would not correct the errors caused by the spiral filter, it would avoid the unwrapping process, which is in many cases a delicate task.

Generally speaking, to achieve better resolution and noise suppression, the multi-image method can incorporate an arbitrary number of interferograms, whose reference phase values are evenly distributed. Obtaining interferograms with different phases of the reference wave is done straightforwardly by shifting the phase of the central circular area of the spiral phase plate, resulting in "revolving" interference spirals. For the case of a non-holographic spi- ral phase element, designed as an on-axis phase plate (as in Fig.l), phase shifting could be performed by just rotating the spiral phase element by the desired phase shifting angle. Compared to an alternative method proposed in Ref. 14, where self-referenced phase-stepping was performed in a phase-contrast microscope setup, the spiral method has the significant advantage that the fringe contrast does not change during stepping of the phase of the ze- roth Fourier order, thus allowing to process a large number of interferograms with smoothly changing fringe positions (and equal fringe contrast) for obtaining highest precision.

3. Discussion

We have investigated a novel Fourier filtering technique which gives an unambiguous impres- sion of the topography of an examined object "at a glance": Using the phase singularity of a vortex filter leads to open interference fringes in the form of spirals with opposite sense of rotation for optical elevations and depressions, respectively. It has been shown theoretically and experimentally that the presented method promises major advantages compared to es- tablished techniques in interferometry, in particular to interference microscopy. Additionally, various demodulation procedures were suggested, based on single- and multiple interferogram demodulation, respectively. Single-image methods are potentially interesting for the examination of very fast processes, for instance for fluid dynamics imaging, because they allow three dimensional topography reconstruction based on a movie recording the interferograms developing in time. The introduced multi-image technique is based on estab- lished demodulation methods and has the advantage of higher accuracy compared to single image demodulation.

4. Appendix

4.1. Derivation of the spiral kernel

The convolution kernel basically is a Fourier transform of the filter function — exp (iθ):¹⁵

where λ the light wavelength and / the focal length of the lens.

The structure of the vortex filter suggests the introduction of polar coordinates:

which lead (after a simplification using a trigonometric sum formula) to

Eq.14 c finally be expressed as

using Eq. 16, which gives an integral representation of the first kind Bessel functions.¹⁶

In its integrated form, the kernel has the form¹

w ere HQ and H_x are Struve functions of zero and first order, respectively. 4..2. Derivation of the vortex filter result

Inserting the first order approximation of Eq.6 into Eq.5 yields

Due to the integration over exp (iφp), the first term equals zero. The following terms can be simplified using the exponential forms of the cos-functions:

Th integrals containing exp (i2φ_P) yield again zero. Integration of the remaining terms finally leads to the form

Here, the scaling factor N has been replaced by {Rl_ut — R^_n) π.

4..3. Multiple image demodulation

The intensity distribution of a general interferogram can be written as

and the related complex image /_c = / exp (iφ_ref) consequently as

The mean value I₀ of three images I₀, which differ only by their reference phase, is

If the p ase values ψ_n are now equally distributed within the interval [0, 2τr], i.e.,

the terms in square brackets give zero, and the expression of Eq.ll remains.

Together with the mean value of the three interferograms

it can finally be shown that th reference amplitude is given by

Acknowledgements

The authors want to thank the Austrian Science Foundation (Project No. P18051-N02) as well as the Austrian Academy of Sciences for support (A. J.).

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16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (New York: Dover, 1970).

5. List of Figure Captions

Fig. 1. Schematic vortex filter setup. The spiral filter is modified such that its central area is assigned a constant phase value. The gray tones of the filter correspond to respective phase shifts. Note that the coordinate system (x', y') is mirrored comp em (x, y). Fig. 2. Comparison of filter kernels K and Ky for /= 0.1m and

(a) and (b) show the absolute values, (c) and (d) the phase of the kernels. Note that the axis scaling has been changed with respect to (a) and (b) for better visualization, (e) and (f) finally show the real parts of the cross section defined by

Fig. 3. Graphical scheme to explain the convolution process for the case of a pure amplitude object: The result at a certain location is derived by shifting the (mirrored) kernel to this point, and integrating over the product of the shifted kernel with Ei_n(x, y). Fig. 4. (a) Gaussian shaped elevation as phase object, (b) After the filter process: A phase factor proportional to the geometrical direction of the local phase gradient has been added. Fig. 5. Demodulation using contour lines: (a) shows the "classical" closed-fringe inter- ferogram of a deformation in a plastic film, (b) the according spiral interferogram. (c) and (d) show a single contour line raw and after preprocessing, respectively. Finally, the reconstructed three-dimensional shape is shown in (e) and (f).

Fig. 6. Demodulation based on center lines: (a) shows a "skeleton" of the spiral fringe pattern, which roughly consists of connected intensity maxima, (b) and (c) show the reconstructed three-dimensional shape.

Fig. 7. Principle of multi-image-demodulation. The mean value of three "complexified" images is proportional to the spiral filtered object field, but without its zeroth order. After restoring the missing field amplitude, the spiral back transformation is accomplished. Finally, the original phase distribution is restored by using a standard phase unwrapping algorithm.

Fig. 1. Schematic vortex filter setup. The spiral filter is modified such that its central area is assigned a constant phase value. The gray tones of the filter correspond to respective phase shifts. Note that the coordinate system (rr', y') is mirrored compared to the system (x, y). Figl.eps.

Fig. 2. Comparison, of filter kernels K and Kv for /= 0.1m and

(a) and (b) show the absolute values, (c) and (d) the phase of the kernels. Note that the axis scaling has been changed with respect to (a) and (b) for better visualization, (e) and (f ) finally show the real parts of the cross section defined by Fig2.eps

Fig. 3. Graphical scheme to explain the convolution process for the case of a pure amplitude object: The result at a certain location is derived by shifting the (mirrored) kernel to this point, and integrating over the product of the shifted kernel with E_m(x, y). Fig3.eps

Fig. 4. (a) Gaussian shaped elevation as phase object, (b) After the filter process: A phase factor proportional to the geometrical direction of the local phase gradient has been added. Fig4.eps.

Pig. 5. Demodulation using contour lines: (a) shows the "classical" closed- fringe interferogram of a deformation in a plastic film, (b) the according spiral interferogram. (c) and (d) show a single contour line raw and after preprocess- ing, respectively. Finally, the reconstructed three-dimensional shape is shown in (e) and (f). Fig5.eps.

Fig. 6. Demodulation based on center lines: (a) shows a "skeleton" of the s fringe pattern, which roughly consists of connected intensity maxima, (b) (c) show the reconstructed three-dimensional shape. Fig6.eps

Fig. 7. Principle of multi-image-demodulation. The mean value of three "com- plexified" images is proportional to the spiral filtered object field, but without its zeroth order. After restoring the missing field amplitude, the spiral back transformation is accomplished. Finally, the original phase distribution is re- stored by using a standard phase unwrapping algorithm. Fig7.eps.

Claims

CLAIMS:

1. A microscopic device comprising: stage means (2) for receiving a sample; a light source (1, 1') for illuminating the sample; optical means (31, 32, 33) for imaging the light transmitted through or reflected by the sample, the optical means (31, 32, 33) defining a light path having at least one focal plane; a detector (4) for detecting the light imaged by the optical means; and a spiral phase element (5) positioned in one of said at least one focal plane of the light path for enhancing the edge contrast of an image detected by the detector (4).

2. The device according to claim 1, wherein said spiral phase element (5) comprises a vortex lens, a blazed phase hologram, or a spatial light modulator.

3. The device according to claim 1 or 2, wherein said spiral phase element (5) is transparent.

4. The device according to any one of the preceding claims, wherein said spiral phase element (5) is arranged substantially perpendicular to the light path.

5. The device according to claim 1 or 2, wherein said spiral phase element (5) is reflective.

6. The device according to any one of the preceding claims, wherein said spiral phase element is centered with respect to the zeroth order Fourier component imaged by the optical means (31 , 32, 33).

7. The device according to any one of the preceding claims, wherein said light source (1, 1') comprises a laser diode or a broadband light source.

8. The device according to any one of the preceding claims, wherein said light source (1, 1') is adapted to emit a plane light wave.

9. The device according to any of claims 1 to 7, wherein said light source has a sufficiently large spatial coherence, such that the light can be focused to a sufficiently small spot in a Fourier plane of the beam path.

10. The device according to any one of the preceding claims, wherein said optical means comprises a microscope objective (31).

11. The device according to claim 10, wherein said optical means further comprises an optical system (32, 33) arranged such that focal plane of the objective (31) is located in the plane the spiral phase element (5) is positioned in.

12. The device according to any one of the preceding claims, wherein the spiral phase element (5) has a spiral shaped phase profile of the form exp(iφ).

13. The device according to any one of the preceding claims, wherein said spiral phase element (5) has a singularity in its center.

14. The device according to any one of claims 1-13, wherein said spiral phase element (5) has an area in its center having a filtering property being a uniform phase shifting operation.

15. A method for enhancing the edge contrast of an image observed with a microscope, in particular according to any one of the preceding claims, comprising the steps of: illuminating a sample; imaging the light transmitted through or reflected by the sample, the imaged light following a light path having at least one focal plane; filtering the light in one of said at least one focal plane of the light path by a spiral phase element (5); and detecting the light transmitted or reflected by said spiral phase element (5).

16. The method according to claim 15, wherein said spiral phase element (5) comprising an area in its center having a filtering property being a uniform phase shifting operation.

17. The method according to claim 16, wherein the light which is transmitted or reflected by said central area interferes with the light transmitted or reflected outside said central area.

18. The method according to claim 16 or 17, wherein an amount of said phase shift in said central area is varied during detection of said image.

19. The method according to claim 16 or 17, wherein the spiral phase element is rotated in its plane around its center.

20. The method according to claim 18 or 19, wherein the method is repeated at least three times for recording at least three images with equivalent phase shift, and the at least three images are numerically processed to obtain a resulting image.

21. The method according to any one of claims 15 to 20, wherein a broadband light source (I¹) with sufficiently high spatial coherence to focus at a sufficiently small spot in the Fourier filter plane is used.

22. The method as in claim 21, where dispersion effects due to diffraction of broadband light at an off-axis spiral phase hologram are compensated by a further diffraction step at an additional grating.

23. The method of claim 21, wherein the additional grating is arranged on to the spiral phase element (5).