. INTRODUCTIONhe properties of a spiral phase element as spatial filter
n imaging applications has been discussed in several re-ent publications.1–7 Placing the filter in a Fourier planef the optical system leads to strong edge contrast en-ancement for amplitude and phase objects similar tohat of the Nomarski, or differential interference contrast,echnique.8 In a recently published paper9 we consideredhe application of a phase vortex to interferometry. We re-orted the observation of spiral-shaped interference pat-erns that appear when thick phase samples are exam-ned. A special feature of such spiral interference patternss that they contain sufficient topographic information tollow the reconstruction of the object’s phase profile. Con-entional methods10 typically require three different in-erference patterns in order to eliminate the ambiguityetween a step “up” or “down” from a given contour lineith respect to its neighbor.The present paper investigates the theoretical back-
round of “spiral interferometry” and suggests methodsor the demodulation of the coiled fringe patterns. On oneand, it is demonstrated how the overall topographic in-ormation of a sample phase profile can be evaluated from
single interferogram. This might have applications inigh-speed interferometry with a single laser pulse, or inhe recording of an interferometric video of a rapidly vary-ng surface in which each video frame can be postpro-essed separately. On the other hand—if imaging speed isot of primary importance—the setup allows high-ccuracy object reconstruction from a selectable numberf interferograms, which in our approach can easily beroduced using a straightforward nonmechanical phase-tepping method.
. VORTEX FILTERINGigure 1 shows the generic experimental setup, which is ao-called 4-f system. The object of interest in the �x ,y�lane is illuminated with sufficiently coherent light. Aonvex lens performs a Fourier transform of the field dis-ribution Ein�x ,y�. After influencing the image by a vortexhase filter, a second Fourier transform is accomplished
y another, identical lens. As indicated in Fig. 1, thehase vortex is slightly modified, i.e., its center is re-laced by a circular area of constant phase shift. The rea-on for this modification is our intention to use the zerothrder spot of the object light field as reference wave forroducing so-called “self-referenced” interferograms (seeef. 9 for details of the setup) that are captured in thelane �x� ,y��. Since the zeroth order light focuses in theenter of the spiral filter, it is now phase-shifted by a con-tant value (determined by the phase of the central circu-ar area) and becomes a plane reference wave after theecond lens, i.e., in the image plane. In our experimentse use the LC-R 3000 reflective-liquid-crystal spatial
ight modulator of HOLOEYE Photonics for generatinghe filter functions. Its small pixel size of 10 �m allows uso match the diameter of the central disk area to the sizef the zeroth order spot.
The following mathematical considerations are con-erned at first only with the effect of pure spiral phase fil-ering, i.e., they neglect the mentioned modification, as-uming the Fourier filter to be a perfect phase spiral. Theffect of the superposition with a phase-controlled planeave, given by the zeroth Fourier order, is considered
ater.According to the Fourier convolution theorem, the re-
ult of the vortex filter process can be derived by a convo-ution of the original object function Ein�x ,y� with theourier transform of the filter:
Eout�x�,y�� = �Ein * KV��x�,y��
=�x=−�
� �y=−�
�
Ein�x,y�KV�x� − x,y� − y�dxdy.
�1�
ne has to use the mirrored filter function exp�i�� whichs identical to −exp�i�� as a result of the inverting prop-rty of the 4-f system. The Fourier transform of −exp�i��s also called the convolution kernel KV. In polar coordi-ates it has the form
006 Optical Society of America
Fos
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Jesacher et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1401
ig. 2. Comparison of filter kernels K and KV for f=0.1 m and �max=0.01 m. (a) and (b) show the absolute values, (c) and (d) the phasef the kernels. Note that the axis scaling for (c) and (d) has been changed with respect to (a) and (b) for better visualization; (e) and (f)how the real parts of the cross section defined by �=−� /2.
ig. 1. Schematic vortex filter setup. The spiral filter is modified such that its central area is assigned a constant phase value. The grayones of the filter correspond to respective phase shifts. Note that the coordinate system �x� ,y�� is mirrored compared with the �x ,y�ystem.
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1402 J. Opt. Soc. Am. A/Vol. 23, No. 6 /June 2006 Jesacher et al.
KV�r,�� =2�
�fi exp�i���
�=0
�max
� J1�2�
�fr�� d�. �2�
1 is the first-order Bessel function of the first kind, � theight wavelength, and � the focal length of the two lenses.he explicit analytical form of KV�r ,�� is derived in Ap-endix A. It is related to the field distribution of theaguerre–Gaussian mode TEM01
* , which is also known asoptical vortex” or “doughnut mode.”11 In the limit �max
�, KV simplifies to i�f exp�i�� / �2�r2�.12
For comparison, Eq. (3) describes the convolution ker-el of the setup shown in Fig. 1 without a vortex filter,ence representing a simple two-lens imaging system:
K�r,�� =2�
�f ��=0
�max
�J0�2�
�fr�� d� =
�max
rJ1�2�
�f�maxr� .
�3�
�r ,�� represents the point-spread function (PSF) of a cir-ular aperture with radius �max. Comparing the convolu-ion kernels of Eq. (2) and (3) (see also Fig. 2), one candentify the main differences to be the different orders ofhe Bessel functions and the vortex phase factor exp�i��,hich causes the vortex kernel KV to be � dependent.his anisotropy is responsible for the spiral interferenceatterns, as will be shown in the following.A descriptive way to achieve the convolution of two
unctions KV�x ,y� and Ein�x ,y� at a certain point P�Px ,Py� is to mirror KV at the origin, then shift it to point, and finally integrate over the product of the shiftedernel with Ein.Consider, for simplicity, an approximated filter kernel
KV�r,�� = �1
Nexp�i�� for R1 � r � R2
0 elsewhere
, �4�
here N is a scaling factor defined as N= �R22−R1
2�� and1, R2 are the inner and the outer radius of a coil aper-
ure, respectively. This assumption allows the result ofhe convolution at point P to be derived as
ig. 3. Graphical scheme to explain the convolution process forerived by shifting the (mirrored) kernel to this point and integr
Eout�P� =1
N��P=0
2� �rP=R1
R2
exp�i�P�Ein�rP,�P� rPdrPd�P,
�5�
here P= �Px ,Py�. Here �rP ,�P� defines a polar coordinateystem with its origin in the center of the kernel (see Fig.) and Ein the input light field expressed in this local co-rdinate system. The convolution integral has to be evalu-ted for every point P of the input light field Ein.For an investigation of the basic effects of such a con-
olution procedure, we expand Ein�xP ,yP�= �Ein�xP ,yP� �exp�iin�xP ,yP� in a Taylor series to first order:
Ein�xP,yP� �Ein�0��exp�iin�0� + exp�iin�0��gAm�0� · rP
+ i�Ein�0��gPh�0� · rP. �6�
ere gAm=��Ein� and gPh=�in describe the amplitudend phase gradient, respectively, of Ein evaluated at point�=kernel center�, which is the same as the origin 0 in
he local coordinate system �xP ,yP�. Together with Eq. (5),he output light field can, after integration (see Appendix) finally be written
Eout�P� � exp�iin�P�gAm�P�exp�i�Am�P�
+ iEin�P�gPh�P�exp�i�Ph�P�. �7�
ere, �Am�P� and �Ph�P� are the polar angles of the corre-ponding gradients (see Fig. 3). Relation (7) allows aualitative examination of the vortex filtering properties:t is apparent that it consists of two terms that describehe effects of amplitude and phase variations of the inputbject on the filter result. The two terms are proportionalo the absolute values of the gradients gAm and gPh re-pectively, which explains the observed strong isotropicmplification of amplitude and phase edges. The factorsxp�i�Ph� and exp�i�Am� can be interpreted as the mani-estation of gradient-dependent geometric phases in theollowing sense: �Ph and �Am were originally geometricngles that indicated the spatial direction of the respec-ive gradients. These factors now appear in the filteredmage no longer as directions, but as additional phase off-
se of a pure amplitude object. The result at a certain location isover the product of the shifted kernel with E �x ,y�.
the caating
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Jesacher et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1403
ets of the image wave at the appropriate positions. Simi-ar to other manifestations of geometric phases,13 thehase offset does not depend on the magnitude of the am-litude or phase gradient, but only on its geometric char-cteristics, one of which in this case is the direction of theeld gradient.A consequence of the factors exp�i�Ph� and exp�i�Am� is
hat the edge amplification becomes anisotropic when in-erfering Eout with a plane reference wave, because theyroduce a phase difference of � between a rising and fall-ng edge of equal orientation. The resulting “shadow ef-ect” gives the impression of a sidewise illumination in ac-ordance with our earlier observations.7 Because of theactor i in relation (7), this pseudo-illumination shows a0° rotation between pure amplitude and phase samples.In the following discussion we assume the object to be a
ure phase sample (with a thickness in the range of thellumination wavelength or larger). Thus we can neglecthe first term in relation (7). In this case, the anisotropyn edge enhancement that emerges when superposing theltered wavefront with an external plane reference waventers the interference fringes: The shape of the fringesepends not only on the phase distribution in of theample, but on the direction of the local phase gradient.wo different points within the sample showing identicalalues for in but unequal values for �Ph lead to differentnterference results. In this manner it becomes clear that
closed isoline of equal phase in that surrounds a localxtremum and would lead to a closed fringe of equalrightness in classical interferometry will cause an openpiral-shaped fringe in spiral interferometry, because theocal phase gradients along this isoline must necessarilyover direction angles ranging from zero to 2�. The rota-ional direction of such a spiral fringe allows us to distin-uish between local maxima and minima.
In the context of interferometry, the weighting of theesulting amplitude by gPh is unwanted because of its in-uence on the fringe positions. However, this effect, in ourxperience, is of little importance, except very close to lo-al extrema and saddle points; moreover it can be com-letely eliminated by using multiple spiral images for ob-ect reconstruction (see Section 3). To get a betternderstanding of how the vortex filter alters phase pro-les, Fig. 4 shows the result of a numerical simulation. Itonsiders a Gaussian-shaped phase sample as the objectf interest (left image). As is apparent in the figure, the
ig. 4. (a) Gaussian-shaped elevation as phase object. (b) After thf the local phase gradient has been added.
ortex-filtered wavefront (right image) looks somehowimilar to the original, but with an “imprinted” phase spi-al.
. DEMODULATION OF SPIRALNTERFEROGRAMShe analysis of conventional interference patterns is usu-lly based on three interferograms, each showing theame object but taken at different values of the phase ofhe reference wave. The images therefore show slightlyhifted fringe patterns, such that together they containhe whole topographic information of the examinedample. Mathematical combination of the three single im-ges yields the object’s surface structure modulo 2�,hich can finally be unwrapped using one of severalhase unwrapping algorithms. Spiral interferograms, inontrast, allow feasible object reconstruction based onnly a single spiral-filtered image. To the best of ournowledge this efficient phase surface reconstruction “atglance” is a novelty in interferometry.Technically, the reconstruction of objects from spiral
ringes can be done in analogy to methods developed fortraditional” interferometry,10 since the interference spi-als can be transformed into normal closed fringe pat-erns by a numerical reverse spiral transform (similar tohe method indicated below in Fig. 8), which can then berocessed in the usual way with high accuracy. However,or resolving the ambiguity between elevations and de-ressions in the object topography, the required comple-entary information can be extracted from an additional
irect processing of the spiral fringe pattern. In the fol-owing we present two alternative one-image demodula-ion methods of spiral fringes that deliver complete quan-itative information about the topography of a specimennd that can be used to provide the missing topographicnformation for supplementing the more precise demodu-ation techniques of traditional interferometry, as pro-osed, e.g., by Larkin.12
. Single-Image Demodulationhe demodulation techniques presented here are basedn the assumption that the filtered wavefront is of theorm Eout�exp�i�in�P�+�Ph�P��, with constant field am-litude. Consequently, the fringe positions reflect the localalues for +� , modulo 2�, which implies that
r process; a phase factor proportional to the geometrical direction
e filte
in Ph
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1404 J. Opt. Soc. Am. A/Vol. 23, No. 6 /June 2006 Jesacher et al.
od�in+�Ph ,2� is constant at positions of maximumringe intensity. The value of this constant can be selectedn the experiment by adjusting the phase of the zerothourier component (center of the spiral phase plate).9
ince gPh is always perpendicular to the tangent of the lo-al spiral fringe, characterized by the angle tan, we ob-ain the relation mod�in+ tan ,2�=C. Setting the arbi-rary constant C to zero, we may write
in = − tan up to multiples of 2�. �8�
caled to length units, this yields
h = − tan
�n�
2�up to multiples of �n�, �9�
here �n is the difference between the refractive indicesf object and surrounding medium and � the illuminationavelength.According to Eq. (9), a basic way to obtain the topo-
raphic information of a pure phase sample from the in-erferogram is to process the spiral from one end to the
ig. 5. Demodulation using contour lines. (a) “Classical” closed-fpiral interferogram, (c) and (d) single contour line raw and afters shown in (e) and (f).
ther, continuously assigning height information h toach point, the height being determined by the angle ofhe spiral’s tangents. In the following, two variants ofingle-image demodulation methods are presented, whichiffer in the way they represent interference fringes byingle lines.
. Contour Line Demodulationquite simple way to obtain the surface profile is based
n processing contour lines. Fig. 5 demonstrates the pro-ess considering a practical example. Figure 5(b) showshe spiral interferogram of a deformation in a transparentlue strip. Phase modulations of such a shape emerge dueo internal stresses where the rigidity of the film is de-reased by local heating. The opposite coiling directions ofhe spirals in the interferogram indicate that the defor-ation consists of an elevation adjacent to a depression.In a first step one single contour line [Fig. 5(c)] must be
onstructed that is a closed line and connects points ofqual intensity in Fig. 5(b). This can be done using stan-ard image processing software. In our case, the software
nterferogram of a deformation in a plastic film, (b) correspondingcessing, respectively. The reconstructed three-dimensional shape
ringe iprepro
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emtGrFoGtlebctt
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Jesacher et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1405
epresents contour lines by an oriented array of L pairs oforresponding �x ,y� vectors (L being the length of theine), which makes their handling quite simple, becausehe height assignment according to Eq. (9) can be carriedut pointwise following the contour line array from its be-inning to its end.
Unfortunately, this process generates a systematic er-or, since after assigning the height information, i.e., afterdding a third “height” dimension, the resulting curve al-ays shows a discontinuity: Starting and ending points of
he numerical processing—which are direct neighbors inhe two-dimensional contour line—are now separated ineight by a step of �n�. This is a consequence of the num-er of clockwise and counterclockwise revolutions, whichecessarily differ by 1 in a closed two-dimensional curve.owever, this error is tolerable in many cases where an
verview of the topography of an extended phase object isequired with a minimum of computational expense. Inddition, the error can be corrected by further processing,.g., by cutting certain parts out of the line, as demon-trated in the example of Fig. 5. Figures 5(d)–5(f) showhe contour plot after smoothing the lines, the recon-tructed shell, and the surface, respectively. As will be ex-lained in Subsection 3.A.2, an alternative way of single-mage demodulation can be devised that avoids thispecific systematic error.
. Center Line Demodulationn the center line demodulation method, the spirals areepresented by curves that follow the intensity maxima.hey are constructed from the spiral interferogram by ap-lying an algorithm that continuously removes pixels
ig. 6. Demodulation based on center lines. (a) “Skeleton” of thaxima, (b) and (c) reconstructed three-dimensional shapes.
rom the spiral boundaries until a “skeleton” remainsFig. 6(a)]. As a result, one obtains a connected line fol-owing the maxima of the spiral fringes that is furtherrocessed similar to the contour line method described inubsection 3.A.1, i.e., the local tangential direction is cal-ulated and transformed into height information. In con-rast to the contour line method, the maximum intensityine avoids the artificial phase jump. However, the calcu-ation of the maximum intensity positions cuts a line intowo parts at possible branching points of the fringes. Afteralculation of the height characteristics for each branchccording to Eq. (9), the two parts are finally reconnected.his procedure ensures a quite accurate reproduction of
he profile [see Fig. 6(b)]. In a final step, interpolation cane used to construct a continuous surface [Fig. 6(c)].In order to compare the still-remaining reconstruction
rrors of the contour line method and the center lineethod, Fig. 7 shows the result of a numerical simulation
hat assumes an input sample consisting of twoaussian-shaped phase deformations. The topographical
econstruction based on contour lines can be taken fromig. 7(b), which also shows the deviation from the originalbject. The standard deviation in this example is 1.18 rad.enerally, the emerging error is influenced by many fac-
ors, such as specific object shape, illumination wave-ength, and the interpolation method used. In the presentxample, the surface data between the contour lines haveeen acquired using cubic spline interpolation. As in theontour line case, Fig. 7(c) illustrates the error analysis ofhe center line method by means of a simulation. Here,he standard deviation was determined to be 0.87 rad.
The center line method thus provides a more accurate
al fringe pattern, which roughly consists of connected intensity
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1406 J. Opt. Soc. Am. A/Vol. 23, No. 6 /June 2006 Jesacher et al.
econstruction of the object phase profile from a single in-erferogram compared with the contour line method de-cribed before. This advantage, however, is at the expensef more computational effort, since a contouring algo-ithm is usually available in software packages, while afringe tracking” algorithm may not be.
. Multi-Image Demodulationf highest surface reconstruction accuracy is desired, aethod based on processing numerous interferograms
an be devised that is based on processing more than onemage, each showing an interferogram captured at a dif-erent value for the phase of the reference wave. The ben-fits include more accurate results and applicability to ob-ects of nonuniform transmission. As will be shown in theollowing, nearly exact results are obtainable by includinghree (or more) different interferograms.
The intensity of a general interference pattern can beescribed as
ere the reference wave parameters Aref and ref are as-umed to be constants. In our case9 the reference wave is
ig. 7. Quantitative error analysis of the single-image-based dewo Gaussian-shaped phase deformations, (b) and (c) show the rpectively, together with their difference from the original phasehe scalings of the phase object graphs.
he zeroth order of the object itself, which ideally forms alane wavefront of uniform intensity in the image planex� ,y��. The reference phase ref is thereby adjustable bydding appropriate phase shifts j to the center region ofhe filter hologram. In the case of an off-axis hologram,he center region is replaced by a blazed grating of adjust-ble phase offset.After acquisition of three interferograms at different
eference phase values j, the reference phase informa-ion is added to the intensity patterns by multiplicationith exp�ij�. The interferogram I1 captured at the refer-
nce phase value 1 thus yields a complex image of theorm Ic1=I1 exp�i1�. The arithmetic mean Ic of the threeomplex images Icj yields
Ic�x�,y�� =1
n j=1
n
Icj = ArefAobj�x�,y��exp�iobj�x�,y��,
�11�
rovided that the reference phase values j are evenlyistributed within the interval �0,2�, that is,
j=1n exp�ij�=0. Ic�x� ,y�� now includes the complete object
opography. The factor Aref can be determined from thehree interferograms; see Appendix A.
ation methods. The input object (a) of the simulation consists ofof the contour line and center line reconstruction methods, re-. Note that the axis scalings of the error plots are different from
modulesults
sample
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Jesacher et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1407
However, strictly speaking, Eq. (11) contains the com-lete object only for interferograms created by a separatexternal reference wave. In the case of self-referenced in-erferometry, we must keep in mind that the object’s ze-oth order itself represents the reference beam. Accordingo Eq. (10), this means that Aobj exp�iobj� describes thebject without its zeroth order. Consequently, Aref has toe added for a complete image reconstruction.So far, the demodulation does not exist in the original
ample, but in its vortex filtered image (see Fig. 8). At thisoint, different ways of further numerical image process-ng are possible. The first one, which is explained in Fig.
ig. 8. Principle of multi-image demodulation. The mean valuef three “complexified” images is proportional to the spiral fil-ered object field, but without its zeroth order. After restorationf the missing field amplitude, the spiral back transformation isccomplished. Finally, the original phase distribution is restoredy using a standard phase unwrapping algorithm.
, consists of an inverse spiral filter process, which meansumerical spatial filtering with the function exp�−i�� inrder to get the true object profile, followed by image un-rapping. It compensates for all errors caused by the spi-
al filter, i.e., the influence of object phase variations onhe amplitude of the light field and vice versa [see Eq.7)].
An alternative way could again make use of contourines: One gray level in the vortex filtered image can behosen to achieve the height deconvolution analog to theingle-image techniques. The complete topography can fi-ally be restored by either repeating this process for ev-ry gray level or by matching the remaining parts of themage to the contour line whose height characteristicsave formerly been acquired. Although this procedureould not correct the errors caused by the spiral filter, itould avoid the unwrapping process, which is in many
ases a delicate task.Generally speaking, to achieve better resolution and
oise suppression, the multi-image method can incorpo-ate an arbitrary number of interferograms whose refer-nce phase values are evenly distributed. Obtaining inter-erograms with different phases of the reference wave isone straightforwardly by shifting the phase of the cen-ral circular area of the spiral phase plate, resulting inrevolving” interference spirals. For the case of a nonholo-raphic spiral phase element designed as an on-axishase plate (as in Fig. 1), phase shifting could be per-ormed merely by rotating the spiral phase element by theesired phase shifting angle. Compared with an alterna-ive method proposed in Ref. 14 where self-referencedhase-stepping was performed in a phase-contrast micro-cope setup, the spiral method has the significant advan-age that the fringe contrast does not change during step-ing of the phase of the zeroth Fourier order, thusllowing one to process a large number of interferogramsith smoothly changing fringe positions (and equal fringe
ontrast) to obtain the highest precision.
. DISCUSSIONe have investigated a Fourier filtering technique that
ives an unambiguous impression of the topography of anxamined object “at a glance.” Using the phase singular-ty of a vortex filter leads to open interference fringes inhe form of spirals with opposite senses of rotation for op-ical elevations and depressions. It has been shown theo-etically and experimentally that the presented methodromises major advantages compared with establishedechniques in interferometry, in particular, interferenceicroscopy.Additionally, various demodulation procedures were
uggested based on single- and multi-interferogram de-odulation. Single-image methods are potentially inter-
sting for the examination of very fast processes, for in-tance, for fluid dynamics imaging, because they allowhree-dimensional topography reconstruction based on aovie recording the interferograms developing in time.he multi-image technique that we have introduced isased on established demodulation methods and has thedvantage of higher accuracy compared with single-imageemodulation.
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1408 J. Opt. Soc. Am. A/Vol. 23, No. 6 /June 2006 Jesacher et al.
PPENDIX A. Derivation of the Spiral Kernelhe convolution kernel basically is a Fourier transform ofhe filter function −exp�i��15:
KV�x,y� = −1
�f � �Aperture
exp�i���,��
exp�− i2�
�f�x� + y���d�d�, �A1�
here � is the light wavelength, f is the focal length of theens, and the aperture is a circularly symmetric disk.
The structure of the vortex filter suggests the introduc-ion of polar coordinates
x = r cos �, y = r sin �, � = � cos �, � = � sin �,
�A2�
hich lead (after a simplification using a trigonometricum formula) to
KV�r,�� = −exp�i��
�f ��=0
�max��=0
2�
� exp�− i��
exp�− i2�
�fr� cos ��d�d�. �A3�
quation (A3) can finally be expressed as
KV�r,�� =2�
�fi exp�i���
�=0
�max
� J1�2�
�fr��d�, �A4�
sing Eq. (A5), which gives an integral representation ofessel functions of the first kind16:
Jn�z� =i−n
2��
�=0
2�
exp�iz cos ��exp�in��d�. �A5�
n its integrated form, the kernel has the form1
KV�r,�� = i exp�i����max
2r �J1�2�
�fr��H0�2�
�fr��
− J0�2�
�fr��H1�2�
�fr��� , �A6�
here H0 and H1 are Struve functions of zero and first or-er, respectively.
. Derivation of the Vortex Filter Resultnserting the first-order approximation of relation (6) intoq. (5) yields
Eout�P� Ein�P�
N ��P=0
2� �rP=R1
R2
exp�i�P�rPdrPd�P
+exp�iin�P�gAm�P�
N �� exp�i�P�rP2
cos��P − �Am�P�drPd�P
+ iEin�P�gPh�P�
N �� exp�i�P�rP2
cos��P − �Ph�P�drPd�P. �A7�
ecause of the integration over exp�i�P�, the first termquals zero. The remaining terms can be simplified usinghe exponential forms of the cosine functions:
out�P� exp�iin�P�gAm�P�
2N �exp�− i�Am�P�
��P
�rP
exp�i2�P�rP2drPd�P
+ exp�i�Am�P���P
�rP
rP2drPd�P�
+ iEin�P�gPh�P�
2N �exp�− i�Ph�P�
��P
�rP
exp�i2�P�rP2drPd�P
+ exp�i�Ph�P���P
�rP
rP2drPd�P� . �A8�
he integrals containing exp�i2�P� again yield zero. Inte-ration of the remaining terms finally leads to the form
Eout�P� 1
3
R23 − R1
3
R22 − R1
2 �exp�iin�P�gAm�P�exp�i�Am�P�
+ iEin�P�gPh�P�exp�i�Ph�P��. �A9�
ere, the scaling factor N has been replaced by �R22
R12��.
. Multiple-Image Demodulationhe intensity distribution of a general interferogram cane written as
I = �Aobj exp�iobj� + Aref exp�iref��2
= Aobj2 + Aref
2 + AobjAref�exp�i�obj − ref�
+ exp�− i�obj − ref��, �A10�
nd the related complex image Ic=I exp�iref� conse-uently as
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1
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Jesacher et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1409
Ic = �Aobj2 + Aref
2 �exp�iref� + AobjAref exp�iobj�
+ AobjAref exp�− iobj�exp�i2ref�. �A11�
he mean value Ic of three images Ic that differ only byheir reference phase is
Ic =1
3�Aobj
2 + Aref2 ��exp�i1� + exp�i2� + exp�i3�
+ AobjAref exp�iobj� +1
3AobjAref exp�− iobj��exp�i21�
+ exp�i22� + exp�i23�. �A12�
f the phase values n are now equally distributed withinhe interval �0,2�, i.e., j=1
n exp�ij�=0, the terms inquare brackets give zero, and the expression of Eq. (11)esults.
Together with the mean value of the three interfero-rams,
I =1
3 n
In = Aobj2 + Aref
2 , �A13�
t can finally be shown that the reference amplitude isiven by
Aref = �I/2 ± ��I/2�2 − �Ic�21/2. �A14�
ote the ambiguity of Aref due to the sign of the innerquare root.
CKNOWLEDGMENTShe authors thank the Austrian Science Foundation
project P18051-N02) as well as the Austrian Academy ofciences for support (A. J.).
Corresponding author S. Bernet’s e-mail address is
EFERENCES1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer,
and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt.39, 1147–1154 (1992).
2. Z. Jaroszewicz and A. Kolodziejczyk, “Zone platesperforming generalized Hankel transforms and theirmetrological applications,” Opt. Commun. 102, 391–396(1993).
3. G. A. Swartzlander, “Peering into darkness with a vortexspatial filter,” Opt. Lett. 26, 497–499 (2001).
4. K. Crabtree, J. A. Davis, and I. Moreno, “Optical processingwith vortex-producing lenses,” Appl. Opt. 43, 1360–1367(2004).
5. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J.Campos, “Image processing with the radial Hilberttransform: theory and experiments,” Opt. Lett. 25, 99–101(2000).
6. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte,“Spiral phase contrast imaging in microscopy,” Opt.Express 13, 689–694 (2005).
7. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte,“Shadow effects in spiral phase contrast microscopy,” Phys.Rev. Lett. 94, 233902 (2005).
8. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith,and C. J. Cogswell, “Linear phase imaging usingdifferential interference contrast microscopy,” J. Microsc.(Oxford) 214, 7–12 (2004).
9. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte,“Spiral interferometry,” Opt. Lett. 30, 1953–1955 (2005).
0. D. W. Robinson and G. T. Reid, eds., InterferogramAnalysis: Digital Fringe Pattern Measurement Techniques(IOP, 1993).
1. S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure andscaling of helical modes of light,” Opt. Lett. 30, 477–479(2005).
2. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Naturaldemodulation of two-dimensional fringe patterns. I.General background of the spiral phase quadraturetransform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
3. J. Anandan, J. Christian, and K. Wanelik, “Resource LetterGPP-1: Geometric Phases in Physics,” Am. J. Phys. 65,180–185 (1997).
4. A. Y. M. Ng, C. W. See, and M. G. Somekh, “Quantitativeoptical microscope with enhanced resolution using apixelated liquid crystal spatial light modulator,” J. Microsc.(Oxford) 214, 334–340 (2004).
5. G. O. Reynolds, J. B. Develis, and B. J. Thompson, The NewPhysical Optics Notebook: Tutorials in Fourier Optics(SPIE, 1989).
