Abstract: Recently a spatial spiral phase filter in a Fourier plane ofa microscopic imaging setup has been demonstrated to produce edgeenhancement and relief-like shadow formation of amplitude and phasesamples. Here we demonstrate that a sequence of at least 3 spatially filteredimages, which are recorded with different rotational orientations of thespiral phase plate, can be used to obtain a quantitative reconstruction ofboth, amplitude and phase information of a complex microscopic sample,i.e. an object consisting of mixed absorptive and refractive components. Themethod is demonstrated using a calibrated phase sample, and an epithelialcheek cell.
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1. Introduction
The use of a spiral phase plate as a spatial filter in a Fourier plane of an imaging setup has beenproposed [1, 2, 3] and demonstrated [4, 5, 6] as an isotropic edge detection method providingstrong contrast enhancement of microscopic amplitude and phase samples. A similar imagingprocedure 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 uniqueproperty that a complete sample phase topography can be unambiguously reconstructed fromonly one single interferogram [7, 8].
Recently [9], the experimental significance of the central singularity of the spiral phaseplate has been pointed out. It was shown that the effect of a transmissive central pixel in a spiralphase plate leads to a violation of the otherwise isotropic edge enhancement, resulting in usefulrelief-like shadow images of the sample topography. The shadow orientations can be rotatedcontinuously by shifting the phase of this central pixel with respect to the remaining spiralphase plate. For optically thin samples it was shown [9] that the shadow effect can be used toobtain a high contrast image of a phase sample by numerical post-processing of a sequence ofat 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, themethod provides a quantitative relative measurement of the amplitude transmission of a sample(normalized to its maximum transmission), and even an absolute measurement of the phasetopography without the need of a previous calibration or comparison with a reference sam-ple. Such a quantitative measurement is hard to achieve with other microscopic methods likestandard phase-contrast or differential interference contrast (Nomarski-) methods [10], whichdeliver just qualitative data. The spiral phase method for a quantitative measurement of both, ab-solute optical thickness and transmission of complex samples has various practical applications,like e.g., lithography mask inspection in semiconductor industry, or quantitative measurementsof biological objects.
2. Basics of spiral phase filtering
The significance of the spiral phase transform, which is also known as the Riesz transform, vor-tex transform, or two-dimensional isotropic Hilbert-transform has been pointed out in differentpublications. As a purely numerical tool, the method is used for example in fringe analysis ofinterferograms [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, byintroducing a spiral phase filter into a Fourier plane of an imaging setup. These experiments
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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 program-mable 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 Fourierfilter is shown in Fig. 1.
Fig. 1. Basic principle of a spiral phase plate spatial Fourier filter. A transmissive inputimage is illuminated by a plane wave. The illumination beam is scattered into the directionsof amplitude or phase gradients within the input image (two directions indicated by redand blue rays). The largest part of the illumination light passes without being scattered(green rays). A first lens (L1) located at a focal distance after the input image createsa Fourier transform of the image in its right focal plane, where the spiral phase plate islocated. The design of the spiral phase plate is shown below (grey-values correspond tophase values in a range between 0 and 2π). The undiffracted part of the illumination beam(green) corresponds to the zero-order Fourier component of the image field and focuses inthe center of the phase plate. The diffracted parts of the input field (red and blue) focus atdifferent positions at the spiral phase plate (indicated below), which are determined by theirpropagation directions in front of L1, and thus by the gradient directions within the inputimage. The spiral phase plate adds a phase offset to each off-axis beam. A second lens L2placed at a focal distance behind the spiral phase plate performs a reverse Fourier transformand creates the output image in its right focal plane. There, the zero-order component of theincident light field (green) is again a plane wave, superposing coherently with the remaininglight field. This remaining light field now carries a spatially dependent phase-offset withrespect to the input image, which corresponds to the geometrical angle into which the(amplitude- or phase) gradient of the input image is directed.
Typically, the spiral phase transformation is defined as a multiplication of the Fourier trans-form of an input image with a vortex phase profile, i.e. with exp(iφ), where φ is the polar anglein a plane transverse to the light propagation direction measured from the center of the spiralphase plate. This definition excludes information about one point, i.e. the center of the spiralphase element, where a phase singularity exists. However, if a real spiral phase element is used
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as a Fourier spatial filter, this central position becomes of utmost importance, since it coincideswith the zero-order Fourier component of the input image which typically contains the majoramount of the total light intensity.
In practice, real spiral phase elements have a central point which is no singularity, but has awell-defined amplitude- and phase transmission property. For example, in our case the phaseshifting element is a pixelated spatial light modulator with individually addressable phasevalues for its 1920 x 1200 pixels. Only in the case where a central region (or pixel) with a sizeon the order of the zero-order Fourier component of the input image has no transmission, theresulting spiral phase transform is really isotropic, resulting in an isotropically edge enhancedoutput image.
However, if the central region acts as a transmissive phase shifter, then the rotational symme-try of the spiral phase filter is broken. This can be seen by the fact that an absolute orientationof the plate can be (for example) defined by the radial direction where the phase plate valuescorrespond to the phase value of the central pixel. If such a non-isotropic spiral phase filter withtransmissive center is used as a spatial Fourier filter, then the output image shows a relief-likeshadow 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 inputimage diffracts an incoming illumination beam into a well-defined direction, corresponding tothe gradient direction (see Fig. 1). In the Fourier plane of the imaging setup, each of thesewell-defined scattered beams is focused at a certain position, at a polar angle corresponding tothe direction of the gradient. The effect of the spiral phase plate is then, to add a certain phasevalue to this beam, which also corresponds to the polar angle of the beam position in the Fourierplane, i.e. to the gradient direction in the original image. Afterwards, the light field is Fourierback-transformed by an additional lens into an output image. Compared to the input image, theoutput image has therefore an additional phase offset at the positions where the input image hasan amplitude or phase gradient. These additional phase offsets equal the geometric directionangles into which the gradients within the input image are pointing.
If such an output image is interferometrically superposed with a plane wave, the ”interfero-gram” will differ from the input image at all positions where the sample has an amplitude orphase gradient. There will always be one gradient direction showing maximum constructiveinterference, i.e. edge amplification, whereas the opposite gradient direction shows maximaldestructive interference, i.e. an edge ”shadow”. Image regions where the gradients have otherdirections show a smooth transition between constructive and destructive interference. This be-havior creates useful pseudo-relief shadow images, where elevations and depressions within aphase topography can be distinguished at a glance.
In the case of an non-isotropic spiral phase plate with transmissive center the plane wave re-quired for the interferometric superposition is automatically delivered by the zero-order Fouriercomponent of the input image field, focussing in the center of the spiral phase plate, since sucha focal point is automatically transformed into a plane wave by the reverse Fourier transformperformed by the following lens. Thus, a spiral phase plate with a transmissive center acts effec-tively as a self-referenced (or common-path) interferometer [16, 17, 18], using the unmodulatedzero-order component of the original input light field as a reference wave for interferometricsuperposition with the remaining, modulated image field. Therefore, changing the phase of thecentral pixel of the spiral phase plate results in a corresponding rotation of the apparent shadowdirection. The same effect can be observed, if the phase of the central pixel is kept constant, butthe 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 reconstructthe exact phase and amplitude transmission of a complex sample. Basically, the possibility todistinguish amplitude from phase modulations results from a π/2-phase offset between the scat-
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tering phases of amplitude and phase structures, resulting in a corresponding rotation angle ofπ/2 between the respective shadow orientations [9]. The feature that an image can be uniquelyreconstructed is based on the fact that there is no information loss in non-isotropically spiralphase filtered light fields (using a transmissive center of the spiral phase element) as comparedto the original light fields, due to the reversibility of the spiral phase transform. For exam-ple, a second spiral phase transform with a complimentary spiral phase element (consisting ofcomplex conjugate phase pixels) can reverse the whole transform without any loss in phase oramplitude information. Note that this would not be the case, for example, if a spiral phase filterof the form ρ exp(iφ) (where ρ is the radial polar coordinate) was used. Although such a filterwould produce the true two-dimensional gradient of a sample [19], it simultaneously erasesthe 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 ofan unclear plane wave offset in the output image. This missing information would not just re-sult in an insignificant intensity offset, but in a strong corruption of the image, since the planewave offset coherently superposes with the remaining image field, leading to an amplificationor 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 niceimpression of the sample topography. In contrast to the Nomarski or differential interferencecontrast method [10] - which creates similar shadow images - the spiral phase method worksalso for birefringent samples. For many applications quantitative data about the absolute phaseand transmission topography of a sample are desired. Here we show that such quantitative datacan be obtained by post processing a series of at least three shadow images (even better resultsof real samples are obtained by a higher number of images), recorded at evenly distributedshadow rotation angles in an interval between 0 and 2π .
The intensity distribution of a series of three images Iout1,2,3 = |Eout1,2,3 |2 in the output planeof a spiral phase filtering setup can be written as:
There, Ein = |Ein(x,y)|exp[iθin(x,y)] is the complex amplitude of the input light field,Ein0 = |Ein0 |exp(iθin0) is the constant zero-order Fourier component (including the complexphase) of the input light field, and α1,2,3 are three constant rotation angles of the spiral phaseplate which are adjusted during recording of the three images, and which are evenly distrib-uted in the interval between 0 and 2π , e.g. α1,2,3 = 0,2π/3,4π/3. The symbol ⊗Φ denotesa convolution process with the Fourier transform of the spiral phase plate (i.e. Φ(ρ ,φ) =F−1{exp[iφ(x,y)]} = iexp[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 Fouriercomponent (Ein−Ein0) is convoluted with the reverse Fourier transform of the spiral phase plate(this process corresponds to the actually performed multiplication of the Fourier transform ofthe image field with the spiral phase function [20]), which is rotated during the three expo-sures to three rotational angles α1,2,3. Then the unmodulated zero-order Fourier componentEin0 which has passed through the center of the spiral phase plate is added as a constant planewave. The squared absolute value of these three ”interferograms” corresponds to the intensityimages which are actually recorded.
The three equations (1) can be rewritten as:
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Here, a∗ symbolizes the complex conjugate of a number a. In order to reconstruct the orig-inal image information Ein(x,y), a kind of complex average IC is formed by a numerical mul-tiplication of the three real output images Iout1,2,3 with the three known complex phase factorsexp(−iα1,2,3), and a subsequent averaging:
IC =13[Iout1 exp(−iα1)+ Iout2 exp(−iα2)+ Iout3 exp(−iα3)] (3)
Analyzing this operation, it is obvious that the multiplication with the complex phase factorsexp(−iα1,2,3) supplies the first and the third lines of Eq. (2) with a complex phase angle ofexp(−iα1,2,3), and exp(−2iα1,2,3), respectively, but it cancels the phase term behind the secondline. The subsequent summation over the three complex images leads to a vanishing of all termswith phase factors, since the three angles are evenly distributed within the interval between 0and 2π . Thus the result is:
IC = [(Ein −Ein0)⊗Φ]E∗in0
(4)
Since Ein0 is a (still unknown) constant, the convolution in Eq. (4) can be reversed by numer-ically performing the deconvolution with the inverse convolution function Φ −1, i.e.:
(Ein −Ein0)E∗in0
= IC ⊗Φ−1 (5)
This deconvolution corresponds to a numerical spiral-back transformation, which can be un-ambiguously performed due to the reversibility of the spiral phase transform. In practice, it isdone by a numerical Fourier transform of IC, then a subsequent multiplication with a spiralphase function with the opposite helicity as compared to the experimentally used spiral phaseplate, i.e. with exp[−iφ(x,y)], followed by a reverse Fourier transform. Note that for this nu-merical back-transform it is not necessary to consider the phase value of the central point in thespiral phase kernel, since the zero-order Fourier component of IC is always zero.
Equation (5) suggests that the original image information E in(x,y) can be restored from the”spiral-back-transformed” complex average IC ⊗Φ−1 by:
There, the complex image information Ein(x,y) has been split into its absolute value and itsphase. Therefore, if the intensity |Ein0 |2 of the constant zero-order Fourier component of theinput image is known, it is possible to reconstruct the complete original image information E in
up to an insignificant phase offset θ in0 , which corresponds to the spatially constant phase of thezero-order Fourier component.
Thus, the final task is to calculate the intensity of the zero-order Fourier component of theinput image |Ein0 |2 from the three spiral transformed images. For this purpose, we first calculatethe ”normal” average IAv of the three recorded images, which is an image consisting of real,positive values, i.e.:
IAv =13(Iout1 + Iout2 + Iout3) (7)
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Again, all terms within Eq. (2) which contain a complex phase factor exp(±iα 1,2,3) willvanish after the averaging, due to the fact that the three angles α 1,2,3 are evenly distributedwithin the interval 0 and 2π . The result is:
IAv = |(Ein −Ein0)⊗Φ|2 + |Ein0 |2 (8)
Comparing Eq. (8) with Eq. (4) one obtains
|Ein0 |4 − IAv|Ein0 |2 + |IC|2 = 0, (9)
from which one can calculate the desired value for |Ein0 |2 as:
|Ein0 |2 =12
IAv± 12
√I2Av−4|IC|2 (10)
Note, that using this equation, |Ein0 |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 |2in fact delivers the same value at each image pixel. In practice there can be some jitter due toimage noise around a mean value of |Ein0 |2, which is an indicator for the noise of the imagingsystem and delivers a useful consistency check. In numerical tests and real experiments it turnedout that in this case the best results are obtained by searching the most frequently occurringvalue (rather than the mean value) of |Ein0 |2 in a histogram, and to insert this value for furtherprocessing of Eq. (6).
Interestingly, there are two possible solutions for the intensity |Ein0 |2 of the zero-orderFourier component, which differ by the sign in front of the square root. In the two cases, theconstant intensity of the zero-order Fourier component at each image pixel exceeds or fallsbelow 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 aspatially dependent modulation of the plane ”carrier-wave” by the higher order Fourier compo-nents. This always applies for pure amplitude samples, and for samples with a sufficiently smallphase 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) witha 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 Fouriercomponent is mainly suppressed. In practice, our thin microscopic samples investigated to datehave all been members of the ”low-scattering” group, where the positive sign in front of thesquare root in equation (10) has to be used.
After inserting |Ein0 |2 from equation (10) into Eq. (6), the absolute phase topography of thesample (up to an insignificant offset) is obtained by calculating the complex phase angle of theright 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 previouscalibration. Furthermore, the transmission image of the sample object can be computed by cal-culating the square of the absolute value of the right hand side of Eq. (6). The result correspondsto 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 reducedbackground noise as compared to a standard bright-field image, due to the coherent averagingof IC (see Eq. (3)) over a selectable number of shadow images. There, all image disturbanceswhich are not influenced by the phase shifting during the different exposures (like readout-noiseof the image sensor, stray light, or noise emerging from contaminated optics behind the Fourierplane) are completely suppressed.
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In practice, noise reduction and image quality can be even enhanced by a straightforwardgeneralization of the described method to the imaging of more than three shadow images. Theonly condition for this generalization to multiple exposures is that the rotation angles of thespiral phase plate are evenly spread over the interval between 0 and 2π .
4. Experimental results
Our actual experimental setup for demonstrating the features of the spiral phase transform issketched in Fig. 2, and explained in the figure caption.
Fig. 2. Sketch of the experimental setup: The sample is illuminated with a collimated white-light beam. The transmitted light passes the objective (NA 0.95, 63x), then a first foldingmirror M1, and a set of two lenses L1 and L2, which project the Fourier transform of theimage at the upper part of a reflective SLM. There, a spiral phase creating hologram witha typical fork-like dislocation in its center is displayed (as sketched in the upper part ofthe SLM image). If the zero-order Fourier component of the incident light field coincideswith the central grating dislocation, the first order diffracted light field is the desired spiralphase filtered image, however, with an undesired dispersion due to the bandwidth of theillumination light (indicated as red/green/blue rays in the figure). In order to compensate forthe dispersion, the diffracted light field passes through a further Fourier-transforming lensL3, which creates a real image in its focal plane where a mirror M2 is located. The mirroris adjusted such that the back-reflected light passes again through the Fourier-transforminglens L3 and focuses at another position on the SLM. There, a ”normal” grating with thesame spatial frequency as that of the spiral phase hologram is displayed (lower image at theright side), from where another first-order diffraction process compensates the dispersioninduced by the first one. Finally, the diffracted light field is reflected by a further foldingmirror M3 to a camera objective lens L4, which projects the spatially filtered image at aCCD chip.
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It differs from the principle setup sketched in Fig. 1 in two main points: First, spiral filteringis not performed by an on-axis transmissive spiral phase plate, but instead by diffraction froman off-axis vortex-creating hologram displayed at a high resolution liquid crystal SLM (1920x 1200 pixels, each pixel is 10x10 μm2). Such holographic gratings with a characteristic fork-like dislocation in their center (see upper image at the right side of Fig. 2) are typically usedto 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 generatea sufficiently accurate on-axis spiral phase plate with our spatial light modulator, due to itslimited phase-modulation capabilities. Therefore we use off-axis diffraction, where the phaseof the diffracted light field is encoded with high precision within the spatial arrangement ofthe hologram structures, rather than in the phase shifts of the individual SLM pixels. Thus, thelimited 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 ofthe filtered light field at a second ”normal” grating with the same spatial frequency as usedfor 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 mirrorM2, which are arranged such that a copy of the light field in the upper part of the SLM planeis produced in the lower part of the SLM plane. There, diffraction at a ”normal” second gratingcompensates for the dispersion introduced by the first one, before recording the spiral-phasefiltered image at a CCD camera. This dispersion control would not be necessary, if an on-axisspiral phase plate was used, or in the case of monochromatic illumination.
In order to produce spiral phase filtered images with an adjustable and controlled shadoweffect, a circular area in the central part of the vortex creating hologram with a diameter on theorder of the size of the zero-order Fourier spot of the incident light field (typically 100 micronsdiameter, depending on the collimation of the illumination light, and on the focal lengths ofthe 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 theremaining, spiral-filtered light field. A controlled rotation of the shadow images can then beperformed by shifting the phase of the central grating, or - preferably - by keeping the phase ofthe central grating constant, but rotating the remaining part of the spiral phase hologram (beforecalculating 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-axisspiral 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, correspondingto an optical path difference of (n−1)h≈ 135 nm. Like any pure phase object which is imagedwith an optical system with a limited numerical aperture, there is also some intensity contrastin the image. The mechanism is based on the fact that small phase structures within the objectscatter the transmitted light at diffraction angles which can be larger than the maximal apertureangle of the microscope objective. As a result, sharp changes of the phase structures in an objectappear darker than their unmodulated surroundings. Thus, this ”spurious” intensity contrastmay be reduced by using objectives with a higher numerical aperture, however, for our testexperiment this effect is desired since it provides us with a quasi complex sample (note that theeffect 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 fora quantitative complex sample, since it possesses a structured phase topography as well as an
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intensity modulation.
Fig. 3. Imaging of a Richardson phase pattern. The total length of the two scale bars atthe right and lower parts of the image are 80 μm, each divided into 8 major intervals witha length of 10 μm. All images are displayed as negatives (i.e. dark areas correspond tobright structures in the real images) for better image contrast. (A), (B), and (C) are threeshadow-effect images recorded at spiral phase plate angles of 0, 2π/3 and 4π/3, respec-tively. For comparison, (D) is a brightfield image recorded with our setup by substitutingthe spiral phase hologram at the SLM by a ”normal” grating. (E) and (F) are the corre-sponding intensity and phase images, respectively, obtained by numerical processing of theshadow-images (A)-(C) according to the method described in the text.
Fig. 3(A-C) show three shadow-effect images of the Richardson slide, recorded at three spiralphase plate rotation angles of 0, 2π/3 and 4π/3, respectively. For display purposes, all imagesof 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 waslimited by the diffraction angle of the SLM holograms such that the whole test sample couldnot be recorded at once. Note that no further image processing (like background subtractionetc.) was used. Obviously, the three shadow-effect images produce a relief-like impression ofthe sample topography, similar to Nomarski- or differential interference contrast methods. Forcomparison, Fig. 3(D) shows a bright-field image of the sample, recorded by substituting thespiral phase pattern displayed at the SLM with a ”normal” grating. As mentioned above, in-tensity variations are visible, although the sample is a pure phase-object. The results for theintensity transmission and phase from the numerical processing of the three shadow-effect im-ages are shown in Fig. 3(E) and (F), respectively. As expected, the measured bright-field imagein (D) is in good agreement with its numerically obtained counterpart in (E). Obviously, the bestcontrast 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 inFig. 4 with an accurate surface map of the sample, recorded with an atomic force microscope
(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3801#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006
(AFM, Nanoscope, by courtesy of Michael Helgert, Research Center Carl Zeiss, Jena).
Fig. 4. Comparison of the spiral phase contrast method with data obtained from an atomicforce microscope (AFM). (A) shows a section from the sample displayed in Figure 3(F),which corresponds to the section (B) scanned with the AFM. In (C), the phase topographyof the selected section as measured with the spiral phase method is displayed as a surfaceplot, with the calculated depth of the etched pattern scaled in absolute units. It turns outthat the pattern depth measured with the spiral phase method seems to be (150 ± 20) nm ascompared to the AFM reference measurement, where a depth of (240 ± 10) nm is obtained.
Fig. 4(A) and (B) show the same section of the sample phase profile, as recorded with thespiral phase method (A) and the AFM (B), respectively. The size of the measured area is 25×25μm2. The two images show again a good qualitative agreement, i.e. even details like thepartial damage of the sample in the lower right quadrant of the sample are well reproducedby 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)
(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3802#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006
of the Richardson spiral. Since the thickness of the bars in the outer rings of the star are 3.5μm (outmost ring), 1.75 μm, 1.1 μm, 0.7 μm, and 0.35 μm,respectively, the actual transverseresolution of the spiral filtering method turns out to be on the order of 1 μm. The effect of thislimited spatial resolution becomes clearer in Fig. 4(C), where the phase profile is plotted as asurface 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) decreasesif the spatial resolution comes to its limit, which is the case for structure sizes below 1 μm.However, for larger structures, the contrast and the measured phase profile are independentfrom the shape or the location of the structure, i.e. in Figure 3(F), different etched objects (likethe maple leaves, circles and square, and the Richardson star) show the same depth of the phaseprofile 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 spiralcontrast method (150 ± 20nm) reveals that our method underestimates the optical path lengthdifference by almost 40 %, which disagrees with the theoretical assumption that the spiralphase method should measure absolute phase values even without calibration. More detailedinvestigations show that this underestimation is due to the limited spatial coherence of thewhite-light illumination (emerging from a fiber with a core diameter of 0.4 mm) which is notconsidered in the theoretical investigation of the previous section. Briefly, the limited spatialcoherence of the illumination results in a zero-order Fourier spot in the SLM plane, which isnot diffraction limited but has an extended size, i.e. it is a ”diffraction disc”. All the other Fouriercomponents of the image field in the SLM plane are thus convolved with this disc, resulting ina smeared Fourier transform of the image field in the SLM plane. Such a smearing results ina decrease of the ideal anticipated edge-enhancement (or shadow-) effect. However, since thisedge-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 apparentlysmaller profile depth, which is actually computed.
We tested this assumption by repeating similar measurements with coherent TEM 00 illumi-nation 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 laserillumination is disturbing, since it results in laser speckles. In contrast to the suppression of in-coherent image noise (like background light) by the spiral phase method, the coherent specklenoise is not filtered out.
Nevertheless, the spiral phase method can be used for a quantitative measurement of phasestructures. This is due to the fact that the apparent decrease of the phase profile does not dependon details of the sample, but just on the setup. Therefore, the setup can be calibrated witha reference test sample like the Richardson slide. In our actual setup the results of furthermeasurements 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 cheekcell as recorded by substituting the spiral phase grating at the SLM with a normal gratingis displayed in (A). Image size is 25×25μm2. (B-D) show three corresponding shadow-effectimages recorded with the same settings as used for Fig. 3. Numerical processing of these imagesresults in the calculated intensity transmission (E) and phase profile (F) images. There, detailsof structures within the cell are visible with a high contrast. The phase profile is plotted againin (G) as a surface plot, where the z-axis corresponds to the computed phase profile depth inradians (without consideration of the calibration factor). In order to find the actual optical pathlength difference of structures within the sample, the phase shift at a certain position has tobe multiplied by the corresponding calibration factor of 1.6, and divided by the wavenumber(2π/λ , λ ≈ 570 nm) of the illumination light. For example, the maximal optical path length
(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3803#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006
Fig. 5. Spiral phase imaging of a cheek cell. (A) is a bright-field image of the cell. (B-D)are three shadow-effect images with apparent illumination directions of 0, 2π/3 and 4π/3,respectively. (E) and (F) are numerically processed intensity transmission and phase profileimages of the sample. (G) is a surface plot of the phase profile, displaying the absolutecalculated phase shift (without calibration correction) in radians.
difference between the lower part of the cell (red in Fig. 5) and its surrounding is approximately70 nm. In order to calculate the absolute height of the cell, the difference of the refractive indicesbetween the cell and its surrounding (water) is required. On the other hand, if the actual heightof the cell was measured by another method (e.g. an AFM), then the refractive index of the cellcontents could be determined.
5. Discussion
In this paper we demonstrated a spiral phase contrast method for quantitative imaging of theamplitude transmission and the phase profile of thin, complex samples. The method is based onthe numerical post-processing of a sequence of at least three shadow-effect images, recordedwith different phase offsets between the zero-order Fourier spot, and the remaining, spiral fil-tered part of the image field. After a straightforward numerical algorithm, a complex image isobtained, whose amplitude and phase correspond to the amplitude and phase transmission ofthe imaged object. In principle, the method is supposed to give quantitative phase profiles ofsamples with a height in the sub-wavelength regime, even without requiring a preceding cali-bration. However, measurements performed at a phase sample calibrated with an AFM revealedthat 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 illumi-nation 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 dis-persive elements (like gratings) are in the beam path. For practical imaging, broadband lightillumination is advantageous, since disturbing speckles are suppressed. In this case, the methodcan still be used for quantitative phase measurements, if it is calibrated with a reference phasesample.
If an on-axis spiral phase plate [14] would be used as a transmissive spiral phase filter, un-
(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3804#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006
desired dispersion effects could be avoided without further dispersion control. If such a platewould be implemented in the back aperture plane of a microscope objective, then a rotationof 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 basedon the fact that the setup acts as a self-referenced interferometer, comparing the zero-orderFourier component of a light field with its remainder. Similar self-referenced phase measure-ments can be in principle performed with a ”normal” phase contrast method by stepping thephase of the zero-order Fourier component with respect to the remaining, non-filtered imagefield [16, 17, 18]. However, using a ”normal” phase contrast method, the interference contrastdepends strongly on the phase difference, whereas the spiral phase method ”automatically” de-livers images with a maximal (but spatially rotating) contrast, since the phase of a spiral phasefiltered image always covers the whole range between 0 and 2π . Advantageously, the phaseshifting in the case of spiral phase filtering just requires a rotation of an inserted spiral phaseplate by the desired phase angle, which cannot be achieved as easily with a ”normal” phasecontrast method. In principle, the method can be also used in reflection mode for measuringsurface phase profiles with an expected resolution on the order of 10 nm or better, which canhave applications in material research and semiconductor inspection.
Acknowledgments
The authors want to thank Michael Helgert (Research Center Carl Zeiss, Jena) for the supplyof the Richardson slide, and for the AFM measurements in Fig. 4. This work was supportedby the Austrian Academy of Sciences (A.J.), and by the Austrian Science Foundation (FWF)Project No. P18051-N02.
(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3805#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006
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