## Abstract

Recently a spatial spiral phase filter in a Fourier plane of a microscopic imaging setup has been demonstrated to produce edge enhancement and relief-like shadow formation of amplitude and phase samples. Here we demonstrate that a sequence of at least 3 spatially filtered images, which are recorded with different rotational orientations of the spiral phase plate, can be used to obtain a quantitative reconstruction of both, amplitude and phase information of a complex microscopic sample, i.e. an object consisting of mixed absorptive and refractive components. The method is demonstrated using a calibrated phase sample, and an epithelial cheek cell.

© 2006 Optical Society of America

## 1. Introduction

The use of a spiral phase plate as a spatial filter in a Fourier plane of an imaging setup has been proposed [1, 2, 3] and demonstrated [4, 5, 6] as an isotropic edge detection method providing strong contrast enhancement of microscopic amplitude and phase samples. A similar imaging procedure applied to samples with a larger optical thickness (on the order of a few wavelength) was shown to result in a novel kind of spiral shaped interferograms, which have the unique property that a complete sample phase topography can be unambiguously reconstructed from only one single interferogram [7, 8].

Recently [9], the experimental significance of the central singularity of the spiral phase plate has been pointed out. It was shown that the effect of a transmissive central pixel in a spiral phase plate leads to a violation of the otherwise isotropic edge enhancement, resulting in useful relief-like shadow images of the sample topography. The shadow orientations can be rotated continuously by shifting the phase of this central pixel with respect to the remaining spiral phase plate. For optically thin samples it was shown [9] that the shadow effect can be used to obtain a high contrast image of a phase sample by numerical post-processing of a sequence of at least three spiral-filtered images recorded with different shadow orientations.

Here we demonstrate that this method can be even used for the imaging of a complex sample, i.e. a sample consisting of both, amplitude and refractive index modulations. In principle, the method provides a quantitative *relative* measurement of the amplitude transmission of a sample (normalized to its maximum transmission), and even an *absolute* measurement of the phase topography without the need of a previous calibration or comparison with a reference sample. Such a quantitative measurement is hard to achieve with other microscopic methods like standard phase-contrast or differential interference contrast (Nomarski-) methods [10], which deliver just qualitative data. The spiral phase method for a quantitative measurement of both, absolute optical thickness and transmission of complex samples has various practical applications, like e.g., lithography mask inspection in semiconductor industry, or quantitative measurements of biological objects.

## 2. Basics of spiral phase filtering

The significance of the spiral phase transform, which is also known as the Riesz transform, vortex transform, or two-dimensional isotropic Hilbert-transform has been pointed out in different publications. As a purely numerical tool, the method is used for example in fringe analysis of interferograms [11, 12], or, very recently, as a tool for the analysis of speckle patterns [13].

There are also applications, where the transformation is performed with optical methods, by introducing a spiral phase filter into a Fourier plane of an imaging setup. These experiments have developed rapidly with the availability of high resolution spatial light modulators (SLMs) which can act as two-dimensional arrays of individually addressable pixels, acting as programmable phase shifters. There, the spiral phase transform can be applied with an on-axis element -a so-called spiral phase plate [14], or by diffraction from a specially designed off-axis hologram [15].

A sketch of a so-called 4f-system as one possible setup for implementing a spatial Fourier filter is shown in Fig. 1.

Typically, the spiral phase transformation is defined as a multiplication of the Fourier transform of an input image with a vortex phase profile, i.e. with exp(*iϕ*), where *ϕ* is the polar angle in a plane transverse to the light propagation direction measured from the center of the spiral phase plate. This definition excludes information about one point, i.e. the center of the spiral phase element, where a phase singularity exists. However, if a real spiral phase element is used as a Fourier spatial filter, this central position becomes of utmost importance, since it coincides with the zero-order Fourier component of the input image which typically contains the major amount of the total light intensity.

In practice, real spiral phase elements have a central point which is no singularity, but has a well-defined amplitude- and phase transmission property. For example, in our case the phase shifting element is a pixelated spatial light modulator with individually addressable phase values for its 1920 × 1200 pixels. Only in the case where a central region (or pixel) with a size on the order of the zero-order Fourier component of the input image has no transmission, the resulting spiral phase transform is really isotropic, resulting in an isotropically edge enhanced output image.

However, if the central region acts as a transmissive phase shifter, then the rotational symmetry of the spiral phase filter is broken. This can be seen by the fact that an absolute orientation of the plate can be (for example) defined by the radial direction where the phase plate values correspond to the phase value of the central pixel. If such a non-isotropic spiral phase filter with transmissive center is used as a spatial Fourier filter, then the output image shows a relief-like shadow profile, similar to a topographic surface which is illuminated from an oblique direction.

The reason for this behavior is that each amplitude or phase gradient within the original input image diffracts an incoming illumination beam into a well-defined direction, corresponding to the gradient direction (see Fig. 1). In the Fourier plane of the imaging setup, each of these well-defined scattered beams is focused at a certain position, at a polar angle corresponding to the direction of the gradient. The effect of the spiral phase plate is then, to add a certain phase value to this beam, which also corresponds to the polar angle of the beam position in the Fourier plane, i.e. to the gradient direction in the original image. Afterwards, the light field is Fourier back-transformed by an additional lens into an output image. Compared to the input image, the output image has therefore an additional phase offset at the positions where the input image has an amplitude or phase gradient. These additional phase offsets equal the geometric direction angles into which the gradients within the input image are pointing.

If such an output image is interferometrically superposed with a plane wave, the “interferogram” will differ from the input image at all positions where the sample has an amplitude or phase gradient. There will always be one gradient direction showing maximum constructive interference, i.e. edge amplification, whereas the opposite gradient direction shows maximal destructive interference, i.e. an edge “shadow”. Image regions where the gradients have other directions show a smooth transition between constructive and destructive interference. This behavior creates useful pseudo-relief shadow images, where elevations and depressions within a phase topography can be distinguished at a glance.

In the case of an non-isotropic spiral phase plate with transmissive center the plane wave required for the interferometric superposition is automatically delivered by the zero-order Fourier component of the input image field, focussing in the center of the spiral phase plate, since such a focal point is automatically transformed into a plane wave by the reverse Fourier transform performed by the following lens. Thus, a spiral phase plate with a transmissive center acts effectively as a self-referenced (or common-path) interferometer [16, 17, 18], using the unmodulated zero-order component of the original input light field as a reference wave for interferometric superposition with the remaining, modulated image field. Therefore, changing the phase of the central pixel of the spiral phase plate results in a corresponding rotation of the apparent shadow direction. The same effect can be observed, if the phase of the central pixel is kept constant, but the whole spiral phase plate is rotated by a certain angle around its center.

In the following, it will be shown how this rotating shadow effect can be used to reconstruct the exact phase and amplitude transmission of a complex sample. Basically, the possibility to distinguish amplitude from phase modulations results from *π*/2-phase offset between the scattering phases of amplitude and phase structures, resulting in a corresponding rotation angle of *π*/2 between the respective shadow orientations [9]. The feature that an image can be *uniquely* reconstructed is based on the fact that there is no information loss in *non-isotropically* spiral phase filtered light fields (using a transmissive center of the spiral phase element) as compared to the original light fields, due to the reversibility of the spiral phase transform. For example, a second spiral phase transform with a complimentary spiral phase element (consisting of complex conjugate phase pixels) can reverse the whole transform without any loss in phase or amplitude information. Note that this would not be the case, for example, if a spiral phase filter of the form *ρ* exp(*iϕ*) (where *ρ* is the radial polar coordinate) was used. Although such a filter would produce the true two-dimensional gradient of a sample [19], it simultaneously erases the information about the zero-order Fourier component of a filtered input image. Therefore, the image information could only be restored up to this zero-order information, consisting of an unclear plane wave offset in the output image. This missing information would not just result in an insignificant intensity offset, but in a strong corruption of the image, since the plane wave offset *coherently* superposes with the remaining image field, leading to an amplification or suppression of different components.

## 3. Numerical post-processing of a series of rotated shadow images

The relief-like shadow images obtained from the non-isotropic spiral phase filter give a nice impression of the sample topography. In contrast to the Nomarski or differential interference contrast method [10] - which creates similar shadow images - the spiral phase method works also for birefringent samples. For many applications quantitative data about the absolute phase and transmission topography of a sample are desired. Here we show that such quantitative data can be obtained by post processing a series of at least three shadow images (even better results of real samples are obtained by a higher number of images), recorded at evenly distributed shadow rotation angles in an interval between 0 and 2*π*.

The intensity distribution of a series of three images *I*
_{out1,2,3} = |*E*
_{out1,2,3}|^{2} in the output plane of a spiral phase filtering setup can be written as:

There, *E _{in}* = |

*E*(

_{in}*x*,

*y*)|exp[

*iϕ*(

_{in}*x*,

*y*)] is the

*complex*amplitude of the input light field,

*E*

_{in0}= |

*E*

_{in0}|exp(

*iϕ*

_{in0}) is the constant zero-order Fourier component (including the complex phase) of the input light field, and

*α*

_{1,2,3}are three constant rotation angles of the spiral phase plate which are adjusted during recording of the three images, and which are evenly distributed in the interval between 0 and 2

*π*, e.g.

*α*

_{1,2,3}= 0,2

*π*/3,4

*π*/3. The symbol ⊗Φ denotes a convolution process with the Fourier transform of the spiral phase plate (i.e. Φ(

*ρ*,

*ϕ*) =

*F*

^{-1}{exp[

*iϕ*(

*x*,

*y*)]} =

*i*exp[

*iϕ*(

*x*,

*y*)]/

*ρ*

^{2}, where

*F*

^{-1}symbolizes the reverse Fourier transform [11]).

Thus the three equations (1) mean that the input image field without its zero-order Fourier component (*E _{in}* -

*E*

_{in0}) is convoluted with the reverse Fourier transform of the spiral phase plate (this process corresponds to the actually performed multiplication of the Fourier transform of the image field with the spiral phase function [20]), which is rotated during the three exposures to three rotational angles

*α*

_{1,2,3}. Then the unmodulated zero-order Fourier component

*E*

_{in0}which has passed through the center of the spiral phase plate is added as a constant plane wave. The squared absolute value of these three “interferograms” corresponds to the intensity images which are actually recorded.

The three equations (1) can be rewritten as:

$$\phantom{\rule{5.2em}{0ex}}+\left[\left({E}_{\mathit{in}}-{E}_{{\mathit{in}}_{0}}\right)\otimes \Phi \right]{E}_{{\mathit{in}}_{0}}^{*}\mathrm{exp}\left(i{\alpha}_{\mathrm{1,2,3}}\right)$$

$$\phantom{\rule{6.4em}{0ex}}+{\left[\left({E}_{\mathit{in}}-{E}_{{\mathit{in}}_{0}}\right)\otimes \Phi \right]}^{*}{E}_{{\mathit{in}}_{0}}\mathrm{exp}\left(-i{\alpha}_{\mathrm{1,2,3}}\right)$$

Here, *a*
^{*} symbolizes the complex conjugate of a number *a*. In order to reconstruct the original image information *E _{in}*(

*x*,

*y*), a kind of complex average

*I*is formed by a numerical multiplication of the three real output images

_{C}*I*

_{out1,2,3}with the three known complex phase factors exp(-

*α*

_{1,2,3}), and a subsequent averaging:

Analyzing this operation, it is obvious that the multiplication with the complex phase factors exp(-*iα*
_{1,2,3}) supplies the first and the third lines of Eq. (2) with a complex phase angle of exp(-*iα*
_{1,2,3}), and exp(-2*iα*
_{1,2,3}), respectively, but it cancels the phase term behind the second line. The subsequent summation over the three complex images leads to a vanishing of all terms with phase factors, since the three angles are evenly distributed within the interval between 0 and 2*π*. Thus the result is:

Since *E*
_{in0} is a (still unknown) constant, the convolution in Eq. (4) can be reversed by numerically performing the deconvolution with the inverse convolution function Φ^{-1}, i.e.:

This deconvolution corresponds to a numerical spiral-back transformation, which can be unambiguously performed due to the reversibility of the spiral phase transform. In practice, it is done by a numerical Fourier transform of *I _{C}*, then a subsequent multiplication with a spiral phase function with the opposite helicity as compared to the experimentally used spiral phase plate, i.e. with exp[-

*iϕ*(

*x*,

*y*)], followed by a reverse Fourier transform. Note that for this numerical back-transform it is not necessary to consider the phase value of the central point in the spiral phase kernel, since the zero-order Fourier component of

*I*is always zero.

_{C}Equation (5) suggests that the original image information *E _{in}*(

*x*,

*y*) can be restored from the “spiral-back-transformed” complex average

*I*⊗ Φ

_{C}^{-1}by:

There, the complex image information *E _{in}*(

*x*,

*y*) has been split into its absolute value and its phase. Therefore, if the intensity |

*E*

_{in0}|

^{2}of the constant zero-order Fourier component of the input image is known, it is possible to reconstruct the complete original image information

*E*up to an insignificant phase offset

_{in}*θ*

_{in0}, which corresponds to the spatially constant phase of the zero-order Fourier component.

Thus, the final task is to calculate the intensity of the zero-order Fourier component of the input image |*E*
_{in0}|^{2} from the three spiral transformed images. For this purpose, we first calculate the “normal” average *I _{Av}* of the three recorded images, which is an image consisting of real, positive values, i.e.:

Again, all terms within Eq. (2) which contain a complex phase factor exp(±*iα*
_{1,2,3}) will vanish after the averaging, due to the fact that the three angles *α*
_{1,2,3} are evenly distributed within the interval 0 and 2*π*. The result is:

Comparing Eq. (8) with Eq. (4) one obtains

from which one can calculate the desired value for |*E*
_{in0}|^{2} as:

Note, that using this equation, |*E*
_{in0}|^{2} can be calculated for each image pixel individually, although it should be a constant. For the ideal case of numerically simulated samples, |*E*
_{in0}|^{2} in fact delivers the same value at each image pixel. In practice there can be some jitter due to image noise around a mean value of |*E*
_{in0}|^{2}, which is an indicator for the noise of the imaging system and delivers a useful consistency check. In numerical tests and real experiments it turned out that in this case the best results are obtained by searching the most frequently occurring value (rather than the mean value) of |*E*
_{in0}|^{2} in a histogram, and to insert this value for further processing of Eq. (6).

Interestingly, there are two possible solutions for the intensity |*E*
_{in0}|^{2} of the zero-order Fourier component, which differ by the sign in front of the square root. In the two cases, the constant intensity of the zero-order Fourier component at each image pixel exceeds or falls below one half of the average image intensity. This means, that in the case of a positive sign, most of the total intensity at an image pixel is due to the plane wave contribution of the zero-order Fourier component of the input image, and the actual image information is contained in a spatially dependent modulation of the plane “carrier-wave” by the higher order Fourier components. This always applies for pure amplitude samples, and for samples with a sufficiently small phase modulation, which is typically the case when imaging thin phase objects in microscopy. On the other hand, if the sample has a deep phase modulation (on the order of *π* or larger) with a high spatial frequency then the solution with the negative sign is appropriate. This happens, for example, for strongly scattering samples like ground glass, where the zero-order Fourier component is mainly suppressed. In practice, our thin microscopic samples investigated to date have all been members of the “low-scattering” group, where the positive sign in front of the square root in equation (10) has to be used.

After inserting |*E*
_{in0}|^{2} from equation (10) into Eq. (6), the absolute phase topography of the sample (up to an insignificant offset) is obtained by calculating the complex phase angle of the right hand side of Eq. (6). Note that the sample phase profile is obtained in absolute phase units, i.e. there is no undetermined scaling factor which would have to be determined by a previous calibration. Furthermore, the transmission image of the sample object can be computed by calculating the square of the absolute value of the right hand side of Eq. (6). The result corresponds to a bright-field image of the object, which could be also recorded with a standard microscope. However, the transmission image of the spiral phase filtering method has a strongly reduced background noise as compared to a standard bright-field image, due to the *coherent* averaging of *I _{C}* (see Eq. (3)) over a selectable number of shadow images. There, all image disturbances which are not influenced by the phase shifting during the different exposures (like readout-noise of the image sensor, stray light, or noise emerging from contaminated optics behind the Fourier plane) are completely suppressed.

In practice, noise reduction and image quality can be even enhanced by a straightforward generalization of the described method to the imaging of more than three shadow images. The only condition for this generalization to multiple exposures is that the rotation angles of the spiral phase plate are evenly spread over the interval between 0 and 2*π*.

## 4. Experimental results

Our actual experimental setup for demonstrating the features of the spiral phase transform is sketched in Fig. 2, and explained in the figure caption.

It differs from the principle setup sketched in Fig. 1 in two main points: First, spiral filtering is not performed by an on-axis transmissive spiral phase plate, but instead by diffraction from an off-axis vortex-creating hologram displayed at a high resolution liquid crystal SLM (1920 × 1200 pixels, each pixel is 10×10 *μ*m^{2}). Such holographic gratings with a characteristic fork-like dislocation in their center (see upper image at the right side of Fig. 2) are typically used to create so-called doughnut beams (Laguerre-Gauss modes) from an incident Gaussian beam [15]. The main reason to use diffraction from such a hologram is that we cannot generate a sufficiently accurate on-axis spiral phase plate with our spatial light modulator, due to its limited phase-modulation capabilities. Therefore we use off-axis diffraction, where the phase of the diffracted light field is encoded with high precision within the *spatial* arrangement of the hologram structures, rather than in the phase shifts of the individual SLM pixels. Thus, the limited phase-modulation capabilities of the SLM are only influencing the diffraction efficiency, but not the phase accuracy of the spiral filtered image.

The second difference to the simple principle setup of Fig. 1 is a further diffraction step of the filtered light field at a second “normal” grating with the same spatial frequency as used for the first one, in order to compensate for the dispersion due to the white light illumination. Basically, the setup uses one more Fourier-transforming lens L3, and a back-reflection mirror M2, which are arranged such that a copy of the light field in the upper part of the SLM plane is produced in the lower part of the SLM plane. There, diffraction at a “normal” second grating compensates for the dispersion introduced by the first one, before recording the spiral-phase filtered image at a CCD camera. This dispersion control would not be necessary, if an on-axis spiral phase plate was used, or in the case of monochromatic illumination.

In order to produce spiral phase filtered images with an adjustable and controlled shadow effect, a circular area in the central part of the vortex creating hologram with a diameter on the order of the size of the zero-order Fourier spot of the incident light field (typically 100 microns diameter, depending on the collimation of the illumination light, and on the focal lengths of the objective, and the lens set L1 and L2) is substituted by a “normal” grating. There, the zero-order Fourier component is just deflected (without being filtered) into the same direction as the remaining, spiral-filtered light field. A controlled rotation of the shadow images can then be performed by shifting the phase of the central grating, or - preferably - by keeping the phase of the central grating constant, but rotating the remaining part of the spiral phase hologram (before calculating its superposition with a plane grating in order to produce the off-axis hologram). This second method is the holographic off-axis analogue to a simple rotation of an on-axis spiral phase plate around its center.

This setup was then used for the imaging of a commercially available phase test pattern (so-called “Richardson slide”), which consists of a micro-pattern with a depth on the order of *h* = 240 nm etched into a transmissive silica sample with a refractive index of n=1.56, corresponding to an optical path difference of (*n*-1)*h* ≈ 135 nm. Like any pure phase object which is imaged with an optical system with a limited numerical aperture, there is also some intensity contrast in the image. The mechanism is based on the fact that small phase structures within the object scatter the transmitted light at diffraction angles which can be larger than the maximal aperture angle of the microscope objective. As a result, sharp changes of the phase structures in an object appear darker than their unmodulated surroundings. Thus, this “spurious” intensity contrast may be reduced by using objectives with a higher numerical aperture, however, for our test experiment this effect is desired since it provides us with a quasi complex sample (note that the effect of Fourier filtering does not depend on the mechanism of the intensity modulation, i.e. there is no difference whether a local intensity reduction is due to an absorber in the sample, or to an intensity loss due to scattering). Thus, the Richardson slide can be used as a model for a quantitative complex sample, since it possesses a structured phase topography as well as an intensity modulation.

Fig. 3(A–C) show three shadow-effect images of the Richardson slide, recorded at three spiral phase plate rotation angles of 0, 2*π*/3 and 4*π*/3, respectively. For display purposes, all images of Fig. 3 are printed as negatives, i.e. dark areas correspond to bright ones in the actual images. Each image is assembled by 4 × 2 individual images, since the field of view of the setup was limited by the diffraction angle of the SLM holograms such that the whole test sample could not be recorded at once. Note that no further image processing (like background subtraction etc.) was used. Obviously, the three shadow-effect images produce a relief-like impression of the sample topography, similar to Nomarski- or differential interference contrast methods. For comparison, Fig. 3(D) shows a bright-field image of the sample, recorded by substituting the spiral phase pattern displayed at the SLM with a “normal” grating. As mentioned above, intensity variations are visible, although the sample is a pure phase-object. The results for the intensity transmission and phase from the numerical processing of the three shadow-effect images are shown in Fig. 3(E) and (F), respectively. As expected, the measured bright-field image in (D) is in good agreement with its numerically obtained counterpart in (E). Obviously, the best contrast of the sample is obtained from the phase of the processed image, displayed in image (F). In order to check the accuracy of the method, a section of this phase image is compared in Fig. 4 with an accurate surface map of the sample, recorded with an atomic force microscope (AFM, Nanoscope, by courtesy of Michael Helgert, Research Center Carl Zeiss, Jena).

Fig. 4(A) and (B) show the same section of the sample phase profile, as recorded with the spiral phase method (A) and the AFM (B), respectively. The size of the measured area is 25 × 25*μ*m^{2}. The two images show again a good qualitative agreement, i.e. even details like the partial damage of the sample in the lower right quadrant of the sample are well reproduced by the spiral phase method. However, the resolution of the AFM is obviously much better, i.e. the actual spiral phase image becomes blurred in the third ring (measured from the outside) of the Richardson spiral. Since the thickness of the bars in the outer rings of the star are 3.5 *μ*m (outmost ring), 1.75 *μ*m, 1.1 *μ*m, 0.7 *μ*m, and 0.35 *μ*m, respectively, the actual transverse resolution of the spiral filtering method turns out to be on the order of 1 *μ*m. The effect of this limited spatial resolution becomes clearer in Fig. 4(C), where the phase profile is plotted as a surface plot with absolute etching depth values in nm as obtained by the numerical processing. Obviously, the contrast of the method (i.e. the height of the inner parts of the sample) decreases if the spatial resolution comes to its limit, which is the case for structure sizes below 1 *μ*m. However, for larger structures, the contrast and the measured phase profile are independent from the shape or the location of the structure, i.e. in Figure 3(F), different etched objects (like the maple leaves, circles and square, and the Richardson star) show the same depth of the phase profile within a range of 10%.

However, there is one drawback, concerning the measured absolute depth of the phase profile. The comparison of the groove depths measured by the AFM (240 ± 10 nm) and the spiral contrast method (150 ± 20nm) reveals that our method underestimates the optical path length difference by almost 40 %, which disagrees with the theoretical assumption that the spiral phase method should measure absolute phase values even without calibration. More detailed investigations show that this underestimation is due to the limited spatial coherence of the white-light illumination (emerging from a fiber with a core diameter of 0.4 mm) which is not considered in the theoretical investigation of the previous section. Briefly, the limited spatial coherence of the illumination results in a zero-order Fourier spot in the SLM plane, which is not diffraction limited but has an extended size, i.e. it is a “diffraction disc”. All the other Fourier components of the image field in the SLM plane are thus convolved with this disc, resulting in a smeared Fourier transform of the image field in the SLM plane. Such a smearing results in a decrease of the ideal anticipated edge-enhancement (or shadow-) effect. However, since this edge-enhancement effect encodes the height of a phase profile in the ideal spiral phase method, its reduction due to the limited coherence of the illumination seems to result from an apparently smaller profile depth, which is actually computed.

We tested this assumption by repeating similar measurements with coherent TEM_{00} illumination from a laser diode. There, in fact the structure depth was measured correctly within ± 10% accuracy. However, for practical imaging purposes the *longitudinal* coherence of the laser illumination is disturbing, since it results in laser speckles. In contrast to the suppression of incoherent image noise (like background light) by the spiral phase method, the coherent speckle noise is not filtered out.

Nevertheless, the spiral phase method can be used for a quantitative measurement of phase structures. This is due to the fact that the apparent decrease of the phase profile does not depend on details of the sample, but just on the setup. Therefore, the setup can be calibrated with a reference test sample like the Richardson slide. In our actual setup the results of further measurements can be corrected by considering the 40 % underestimation of the phase depth.

An example for such a measurement is shown in Figure 5. A bright-field image of the cheek cell as recorded by substituting the spiral phase grating at the SLM with a normal grating is displayed in (A). Image size is 25 × 25*μ*m^{2}. (B–D) show three corresponding shadow-effect images recorded with the same settings as used for Fig. 3. Numerical processing of these images results in the calculated intensity transmission (E) and phase profile (F) images. There, details of structures within the cell are visible with a high contrast. The phase profile is plotted again in (G) as a surface plot, where the z-axis corresponds to the computed phase profile depth in radians (without consideration of the calibration factor). In order to find the actual optical path length difference of structures within the sample, the phase shift at a certain position has to be multiplied by the corresponding calibration factor of 1.6, and divided by the wavenumber (2*π*/λ, λ ≈ 570 nm) of the illumination light. For example, the maximal optical path length difference between the lower part of the cell (red in Fig. 5) and its surrounding is approximately 70 nm. In order to calculate the absolute height of the cell, the difference of the refractive indices between the cell and its surrounding (water) is required. On the other hand, if the actual height of the cell was measured by another method (e.g. an AFM), then the refractive index of the cell contents could be determined.

## 5. Discussion

In this paper we demonstrated a spiral phase contrast method for quantitative imaging of the amplitude transmission and the phase profile of thin, complex samples. The method is based on the numerical post-processing of a sequence of at least three shadow-effect images, recorded with different phase offsets between the zero-order Fourier spot, and the remaining, spiral filtered part of the image field. After a straightforward numerical algorithm, a complex image is obtained, whose amplitude and phase correspond to the amplitude and phase transmission of the imaged object. In principle, the method is supposed to give quantitative phase profiles of samples with a height in the sub-wavelength regime, even without requiring a preceding calibration. However, measurements performed at a phase sample calibrated with an AFM revealed that the method underestimates the height of the phase profile. This is due to the limited spatial (transverse) coherence of the illumination system, and could be avoided by using TEM_{00} illumination from a laser diode. On the other hand, there is no requirement for longitudinal (temporal) coherence for performing spiral phase filtering, with the exception of dispersion control if dispersive elements (like gratings) are in the beam path. For practical imaging, broadband light illumination is advantageous, since disturbing speckles are suppressed. In this case, the method can still be used for quantitative phase measurements, if it is calibrated with a reference phase sample.

If an on-axis spiral phase plate [14] would be used as a transmissive spiral phase filter, undesired dispersion effects could be avoided without further dispersion control. If such a plate would be implemented in the back aperture plane of a microscope objective, then a rotation of the shadow direction would just require a corresponding rotation of the spiral phase plate. The actual feature of such a setup to measure sub-wavelength optical path differences is based on the fact that the setup acts as a self-referenced interferometer, comparing the zero-order Fourier component of a light field with its remainder. Similar self-referenced phase measurements can be in principle performed with a “normal” phase contrast method by stepping the phase of the zero-order Fourier component with respect to the remaining, non-filtered image field [16, 17, 18]. However, using a “normal” phase contrast method, the interference contrast depends strongly on the phase difference, whereas the spiral phase method “automatically” delivers images with a maximal (but spatially rotating) contrast, since the phase of a spiral phase filtered image always covers the whole range between 0 and 2*π*. Advantageously, the phase shifting in the case of spiral phase filtering just requires a rotation of an inserted spiral phase plate by the desired phase angle, which cannot be achieved as easily with a “normal” phase contrast method. In principle, the method can be also used in reflection mode for measuring surface phase profiles with an expected resolution on the order of 10 nm or better, which can have applications in material research and semiconductor inspection.

## Acknowledgments

The authors want to thank Michael Helgert (Research Center Carl Zeiss, Jena) for the supply of the Richardson slide, and for the AFM measurements in Fig. 4. This work was supported by the Austrian Academy of Sciences (A.J.), and by the Austrian Science Foundation (FWF) Project No. P18051-N02.

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