Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination

Analysis of systematic errors in spatial carrier phaseshifting applied to interferogram intensity modulation

determination

Adam Styk* and Krzysztof PatorskiInstitute of Micromechanics and Photonics, Warsaw University of Technology, 8 Sw. A. Boboli Street, 02-525 Warsaw, Poland

*Corresponding author: [email protected]

Received 22 February 2007; accepted 26 March 2007;posted 12 April 2007 (Doc. ID 80245); published 6 July 2007

Two-beam interferogram intensity modulation decoding using spatial carrier phase shifting interferometryis discussed. Single frame recording, simplicity of experimental equipment, and uncomplicated data pro-cessing are the main advantages of the method. A comprehensive analysis of the influence of systematicerrors (spatial carrier miscalibration, nonuniform average intensity profile, and nonlinear recording) on themodulation distribution determination using automatic fringe pattern analysis techniques is presented. Theresults of searching for the optimum calculation algorithm are described. Extensive numerical simulationsare compared with laboratory findings obtained when testing vibrating silicon microelements under variousexperimental conditions. © 2007 Optical Society of America

OCIS codes: 120.2650, 120.3180, 120.5060, 120.7280.

1. Introduction

Laser interferometry is a powerful method for accuratequantitative studies at micrometer and nanometerscales. High versatility of interferometric methods of-fers possibilities of applications in diverse physical andengineering fields. Usually, desired information is en-coded in the interferogram phase distribution, i.e.,shape and position of fringes. Quantitative analysis ofinterferograms is performed using automatic fringepattern analysis methods, see, for example, [1]. Alter-natively the required information can be encoded inthe interferogram contrast or intensity modulation dis-tribution. Time average interferometry for vibrationinvestigation, white light interferometry for surfaceprofile and microprofile measurement, and optical co-herence tomography for biotissue studies can serve asexamples.

In the recent works [2,3] extensive studies of theinfluence of major measurement errors of the tempo-ral phase shifting (TPS) technique on the evaluationof the interferogram contrast or modulation distribu-tions have been presented [the contrast and modula-tion terms are defined in the explanation of Eq. (1)].

TPS is the most accurate method for phase andcontrast�modulation studies providing strict mea-surement conditions are maintained. One drawbackof the TPS technique is the necessity of acquiring sev-eral frames. Any environmental disturbances duringthe measurement process cause errors in evaluateddata. Thus single frame recording is an attractive so-lution. The spatial carrier phase shifting technique(SCPS), another popular phase evaluation method,combines one frame recording with TPS data process-ing simplicity.

Since the interferogram modulation determinationis less sensitive to measurement errors than contrastevaluation [3], and the algorithms used in SCPS aremathematically equivalent to the TPS ones (differentdomain is used only) [4], the authors have decided touse the SCPS technique to calculate the modulationdistribution. In this work properties of the SCPSmethod applied to the fringe pattern modulation vi-sualization are analyzed. Detailed studies of the Fou-rier transform method applied for the same purposewere presented in [5].

Numerical simulations of main SCPS experimentalerrors [6], i.e., linear and nonlinear miscalibration ofthe carrier frequency used, nonuniform distribution ofinterferogram average intensity (caused by spatial in-tensity variations of incident wavefronts), and detector

0003-6935/07/214613-12$15.00/0© 2007 Optical Society of America

20 July 2007 � Vol. 46, No. 21 � APPLIED OPTICS 4613

nonlinearities will be presented. Parallel and perpen-dicular directions of two-beam interference fringeswith respect to the direction of interferogram modula-tion changes are considered. To emphasize the influ-ence of simulated measurement errors on theevaluated modulation distribution the influence ofhigh frequency additive intensity noise has been omit-ted.

Experimental work aims at the visualization of res-onant modes of silicon atomic force microscope (AFM)cantilevers under various experimental conditions. Incase of sinusoidal vibrations the interferogram modu-lation distribution is described by the zero order Besselfunction. Numerical and experimental studies arecompared.

2. Spatial Carrier Phase Shifting Algorithms

The intensity distribution in the two-beam interfero-gram can be described as

I�x, y� � I0�x, y��1 � K�x, y��x, y�cos��x, y�� x, y��, (1)

where I0�x, y� is the interferogram bias �DC�; K�x, y�is the contrast envelope function; ��x, y� is the con-trast (normalized visibility) of two-beam interferencefringes; ��x, y� � �2��OPD�x, y�, where OPD�x, y� isthe optical path difference between the referencebeam and object beam wave fronts; ��x, y� is a knownphase shift between test and reference wavefronts(e.g., corresponding to the spatial carrier frequency).The interferogram modulation distribution (the con-trast multiplied by the interferogram bias distribu-tion) depends on three terms: I0�x, y�, K�x, y�, and��x, y�.

The contrast envelope K�x, y� can be described bydifferent functions. For example, in case of time av-erage interferometry applied to sinusoidal vibrationtesting the two-beam interferogram contrast enve-lope is described by |J0�ka0�|, the zero order Besselfunction with a vibration amplitude a0�x, y� encodedin its argument. In this application the influence ofinhomogeneous distribution of the interferogram biasI0�x, y� should be suppressed. The solution of Petit-grand et al. [7] is to find the values of interferogramintensity modulation for vibration state and normal-ize them by the modulation calculated for the staticobject. In result the contrast envelope function|J0�ka0�| can be obtained.

In this paper the properties of the SCPS methodapplied to interferogram modulation distribution cal-culations without normalization process will be stud-ied. They are sufficient to comprehend and explainthe error sources and their influence. A modifiedthree point algorithm and two five point algorithmswill be considered. The three point [8] and one of thefive point [9] algorithms are the examples of the al-gorithms with a high tolerance for phase step error(spatial carrier miscalibration). Its resistance to thephase step error is based on the assumption of theconstant phase derivative in the calculating window.

The considered algorithms are much simpler thanthe algorithms with a higher number of requiredsampling points [10,11]. In the work the four pointalgorithms were intentionally omitted due to an evennumber of pixels in the calculating window. Thisproperty causes substantial and chaotic error distri-bution while calculating modulation distributionsthan the algorithms presented below.

The algorithms used are as follows:

(a) modulation algorithm 3Amod (modified threepoint algorithm [8]):

3Amod: 12��I�2 � I�4�2 � 4I3

� 2� �I�2 � I�4�2, (2)

where

I�i � Ii � 0.5�Ii�1 � Ii�1�, i � 2, 3, 4;

(b) five point modulation algorithm 5Amod [12,13]:

5Amod: 14�4�I2 � I4�2 � �2I3 � I1 � I5�2; (3)

(c) Larkin five point algorithm 5Lmod [9]:

5Lmod: 12��I2 � I4�2 � �I1 � I3��I3 � I5�, (4)

where I1 to I5 denotes intensity values at consecutivepixels in the interferogram. All above N-point algo-rithms [6] use N adjacent pixels with known phaseshifts. The phase shift between pixels is generated byadding tilt fringes (carrier frequency) across the in-terferogram. Algorithms expressed by Eqs. (3) and (4)require a 90° phase shift between consecutive pixelsso each fringe period must cover four pixels. Forphase calculations, algorithm 3Amod uses the phaseshift equal to 120° (each fringe must cover three pix-els); whereas for the modulation distribution evalu-ation the shift of 90° is used. The algorithm 3Amod, Eq.(2), was derived by taking the square root of the sumof squares of the numerator and denominator of thephase calculating algorithm [8]. The numerator and

Fig. 1. Simulated exemplary fringe patterns with Gaussianchanges of the intensity modulation distribution (a) perpendicularand (b) parallel mutual orientation with respect to the interfero-gram carrier fringes.

4614 APPLIED OPTICS � Vol. 46, No. 21 � 20 July 2007

denominator contain an additional term dependingon the phase step angle between adjacent pixels. Itcancels in the phase calculation algorithm, but in themodulation evaluation one, its contribution is signif-icant. With the 90° phase shift between adjacent pix-els it gives a correct modulation evaluation. Thedeparture from the assumed carrier frequency causesan error in evaluated quantities. According to Eq. (1),to keep a constant phase step value between the pix-els, an unknown interferogram phase ��x, y� alsoneeds to be constant across the interferogram. A sim-ilar remark concerns the modulation distribution.N-point algorithms require constant modulation dis-tribution in pixels. A nonuniform interferogrammodulation distribution will lead to an error in the

evaluated modulation maps. Correspondingly, theproperties of the N-point algorithms need to be de-termined.

The error analysis of selected SCPS algorithms ap-plied to the interferogram intensity modulation re-trieval will be focused on the main errors that mightaccompany the measurement process:

Y modulation distribution nonuniformity,Y spatial carrier miscalibrations (linear, non-

linear),Y nonlinear recording.

Analyses are based on extensive numerical simula-tions using MATLAB software.

Fig. 2. (Color online) Simulated interferogram contrast envelope functions across the interferograms.

Fig. 3. (Color online) Differences between the error-free and calculated modulations using algorithms 3Amod, 5Amod, and 5Lmod. Thecontrast envelope was described by linear function, first row; Gaussian function, second row; and Bessel function J0, third row. The caseof perpendicular orientation between the direction of interferogram carrier fringes and modulation changes.


3. Numerical Simulations of Selected Errors

The parameters of numerically generated two-beaminterference fringe patterns were as follows:

1. Fringe pattern dimensions: 256 256 pixels.2. Spatial carrier (sampling rate): variable, from 2

to 32 pixels per fringe.3. Mutual direction of the spatial carrier interfer-

ence fringes and the modulation changes: parallel,perpendicular.

4. Intensity levels: 256 intensity levels, gray scale(8 bit coding).

5. Interferogram average intensity changes: Gauss-ian (the value of the bias at the interferogram edge isset to 10% of its maximum value).

6. Contrast envelope function: linear (valuesrange: 0–1 over the simulated fringe pattern), Gauss-ian (values range: 0–1 over the simulated fringe pat-tern), zero order Bessel function J0 (values range: 0–1over the simulated fringe pattern; Bessel functionargument range: 0–2.5�).

Figure 1 shows exemplary simulated fringe patternswith the intensity modulation changes according toa Gaussian function. The direction of modulationchanges is perpendicular and parallel to the interfero-gram carrier fringes. All numerical simulations wereperformed under the assumption that the contrast��x, y� of two-beam interference fringes is constantover the whole interferogram and equal to 0.7. Thisvalue has been chosen as the one corresponding toexperimental conditions ensuring a sinusoidal in-tensity distribution in the recorded interferogram

(avoiding nonlinear recording). During the experimentthe above assumption might be violated, e.g., when thesample reflectivity varies over the field of view. In thatcase the evaluated modulation distribution is addition-ally modified by the contrast ��x, y� changes, see Eq.(1). When the time average interferometry techniquefor vibration measurements is considered, the abovementioned error can be removed by normalizing theobject vibration state by the static one.

A. Influence of Interferogram Bias and ContrastVariations

As mentioned above the intensity modulation in Eq.(1) is a product of three terms: the contrast envelopefunction K�x, y�, interferogram contrast ��x, y�, andinterferogram bias I0�x, y�. As the spatial carrierphase shifting algorithms are the same as the onesused in temporal phase shifting, every limitation en-countered in the latter technique should be present inthe former one. One of them is an assumption ofconstant modulation at each interferogram point inadjacent frames. This condition cannot be preservedin the spatial carrier technique applied to modulationdistribution determination when the contrast enve-lope function differs from a uniform distribution, be-sides the case when the direction of the spatial carrierfringes is set parallel to the contrast envelope func-tion changes. In this section we present numericalsimulations of the influence of the contrast envelopefunction and the interferogram bias distribution onthe interferogram modulation distribution across theinterferogram.

Fig. 4. (Color online) Difference between the error-free modulation and modulation maps calculated using algorithms 3Amod, 5Amod, and5Lmod. The contrast envelope was described by a Gaussian function. The case of parallel mutual orientation between the interferogramcarrier fringes and the modulation change direction.

Fig. 5. (Color online) Difference between the error-free modulation distribution and the modulation calculated using algorithms 3Amod,5Amod, and 5Lmod. The interferogram bias distribution described by a Gaussian function; perpendicular orientation between the interfero-gram carrier fringes and interferogram bias change direction.


1. Contrast Envelope FunctionWhen simulating the error caused by the contrastenvelope function, two other modulation components,i.e., I0�x, y� and ��x, y� were set constant over the in-terferogram. As the contrast envelope depends on therealized interferometric technique, it was described bylinear, Gaussian, and Bessel functions. Parallel andperpendicular mutual orientations between simulatedrectilinear fringes and their intensity modulationchange direction were considered. To simplify andorganize conclusions the modulation (or contrast)changes were assumed in one direction only. Figure 2

presents three simulated contrast envelope functions.The X axis represents the pixel number while the Yaxis shows normalized functions under investigation.

Figure 3 shows the error distributions (the differ-ence between the error-free and evaluated modula-tion distributions) for three considered algorithms inthe case of three different contrast envelope func-tions. The direction of the modulation changes is per-pendicular to the interferogram carrier fringes [seeFig. 1(a)]. The ordinates are expressed in arbitraryunits (intensity levels), the abscissa correspond topixel numbers.

Fig. 6. (Color online) Difference between the error-free modulation and modulation distributions calculated using three algorithmsconsidered with the presence of the carrier frequency miscalibration. Phase shift between consecutive pixels is equal to 60°, first row, and120°, second row. Uniform modulation distribution was assumed.

Fig. 7. (Color online) Cross sections through the difference between the error-free modulation and the evaluated modulation distributionscalculated using three algorithms under consideration. Phase shift between consecutive pixels: 60°, first row; 120°, second row. Modulationdistribution changes described by the Bessel function J0. The case of perpendicular mutual orientation between interferogram carrierfringes and the interferogram modulation change direction.


If the interferogram fringes run parallel to themodulation change direction [Fig. 1(b)] there is noerror caused by the contrast envelope function. Fig-ure 4 presents the difference between the error-freemodulation and calculated modulation distributionsfor a parallel orientation of the carrier fringes and themodulation distribution direction. The latter is de-scribed by a Gaussian function [Fig. 2(b)] for all con-sidered algorithms. Irregular noise distribution iscaused by intensity digitization. This error distribu-tion changes with a change of initial phase of carrierfringes.

A general conclusion that follows from numericalexperiments is that the changes of the modulationvalues in adjacent pixels generate erroneous re-trieved modulation distribution. The error is mini-mized for the parallel mutual orientation betweenthe interferogram carrier fringes and the modulationdistribution direction. The contrast envelope functionchanges lead to a high-frequency modulation errorat twice the carrier fringe frequency. The error isasymmetrical with respect to 0 and its averagevalue distribution along the modulation changesdiffers for various contrast envelope functions. Al-gorithms 3Amod and 5Lmod give similar error dis-tributions while algorithm 5Amod evaluates themodulation with different error distributions andhigher accuracy.

2. Interferogram Bias Distribution NonuniformityIt is nearly impossible to have a uniform bias distri-bution across the entire image plane. Commonly usedSCPS algorithms assume the same value for all sam-ples. Unfortunately, one of the terms describing the

interferogram modulation distribution is the biasI0�x, y�, see Eq. (1). Below the influence of a nonuni-form interferogram bias distribution on the modula-tion evaluation is discussed. The interferogram biaschanges were assumed to be described by a Gaussianfunction. To highlight the error, the value of the biasat the interferogram edge is set to 10% of its maxi-mum value. The bias change direction is perpendic-ular to the interferogram carrier fringes direction.The contrast envelope function is set to be constantover the whole interferogram and equal to 1. Figure 5presents the difference between the calculated mod-ulation distribution and the error-free one for threeconsidered algorithms.

The interferogram bias changes in adjacent pix-els generate parasitic fringes (error distributionpattern) over the evaluated modulation distribu-tion. The parasitic modulation spatial frequency isequal to two-beam interferogram frequency. Algo-rithms 5Amod and 5Lmod provide similar error distri-butions. The smallest amplitude of the error is inthe area where the bias distribution is nearly con-stant (at the top of the Gaussian function). Algo-rithm 3Amod evaluates the modulation distributionwith higher accuracy.

For the parallel orientation between the interfero-gram carrier fringes and the interferogram biaschange direction, similar results are obtained as inthe case of a parallel orientation of the interferogramcarrier fringes and the contrast envelope function(see Subsection 3.A.1). Noise appearing in calculatedmodulations is caused by intensity digitization (seeFig. 4).

Fig. 8. (Color online) Calibration curves showing change of modulation values evaluated when different spatial carrier frequency isapplied (spatial frequency is rescaled to the relative phase shift between adjacent pixels).

Fig. 9. (Color online) Cross sections through the difference between the error-free modulation and modulation distributions calculatedusing algorithms 3Amod, 5Amod, and 5Lmod showing the influence of a nonlinear carrier frequency miscalibration. See text for details.


B. Influence of the Spatial Carrier Miscalibration

Each SCPS algorithm used for phase evaluation re-quires appropriate spatial carrier frequency. Any de-parture from this condition causes an error in theretrieved phase distribution. The same remark con-cerns the use of SCPS algorithms for modulationevaluation. Algorithms under consideration require a90° phase change between consecutive pixels. In thiswork we have considered the influence of the spatialcarrier frequency miscalibration in the range from60° to 120° (from three to six pixels for one interfer-ence fringe). A uniform distribution and the onesdescribed by different contrast envelopes (linear,Gaussian, Bessel; see Fig. 2) were used to simulatethe modulation distribution in the interferogram.The interferogram bias distribution was set constant.Parallel and perpendicular mutual orientations be-tween interferogram carrier fringes and modulationchange directions were studied.

The difference between the error-free and calcu-lated modulation distributions for the miscalibrationerror and uniform modulation distribution for threealgorithms under consideration is presented in Fig.6. The results obtained using algorithms 3Amod and5Lmod are comparable with respect to the shape (thevalue of the error differs), while the algorithm 5Lmodgives a disparate error distribution (compare withFig. 13). It is worth noticing that the algorithm 3Amodgives a different error sign when different miscalibra-tion values are encountered (see Fig. 8). The errorsign in case of two other algorithms remains un-changed with respect to the miscalibration values.

Subtracted error-free and calculated modulationswith the presence of the carrier frequency miscalibra-tion and the modulation distribution according to theBessel function J0 are presented in Fig. 7. The argu-ment of the Bessel function is in the range of 0–5��2divided into 256 equidistant samples; the cross sec-tion is taken in the direction of modulation changes.The error appearing over the evaluated modulationmaps has the same properties as described in the caseof the uniform modulation distribution and spatialmiscalibration. The magnitude of the error appearingdue to the nonuniformity of the contrast envelopefunction (see Subsection 3.A.1) is far smaller than theone caused by spatial carrier miscalibration thus isunnoticeable in evaluated maps.

For either parallel or perpendicular mutual direc-tions between the modulation distribution changesand the spatial carrier fringes the conclusions con-cerning the influence of the spatial carrier miscali-bration are identical. Algorithms 3Amod and 5Lmod

give smooth reproduction of the modulation distribu-tion (Fig. 6) since they were developed for constantphase shifts between adjacent pixels. The values ofthe calculated modulation distribution at each in-terferogram pixel differ from the error-free one. Ageneral character of the modulation distribution,however, remains. It can be stated that the carrierfrequency miscalibration error causes a “scaling ef-fect” of the determined modulation distribution withrespect to the error-free one. The value of every eval-uated point of the modulation map differs from theerror-free one by the same amount, thus the influence

Fig. 10. (Color online) Error distribution in the interferogram intensity modulation evaluated using algorithms 3Amod, 5Amod, and 5Lmod

in the presence of a nonlinear recording error. Modulation changes according to Bessel function J0; the case of interferogram carrier fringesrunning perpendicularly to the interferogram modulation change direction.

Fig. 11. (Color online) Horizontal cross sections through the modulation maps calculated using considered algorithms in the presence ofnonlinear recording. Modulation changes according to the Bessel function J0. The case of parallel mutual orientation between interfero-gram carrier fringes and the interferogram modulation change direction.


of described error might be canceled by a normaliza-tion process. The percentage amount of evaluatedmodulation value with respect to the error-free onecaused by the spatial carrier miscalibration error ispresented in Fig. 8. The X axis denotes the relativephase shift between adjacent pixels in applied carrierfringes.

If we calculate the modulation distribution in thedirection of its changes, the error caused by the con-trast envelope function (see Subsection 3.A.1) arises.This error is much smaller compared with the errorcaused by the carrier frequency miscalibration, thusit can be omitted.

All above conclusions concern the algorithm 5Amodas well. Additionally, using this algorithm, the car-rier frequency miscalibration generates parasiticfringes in the evaluated modulation map. These par-asitic modulations follow the carrier fringes at a dou-ble frequency. If the carrier frequency miscalibrationerror is relatively small (few fringes in the wholeinterferogram), the parasitic ripples are modulatedby an envelope at the frequency four times the errorfrequency (four ripples for each fringe miscalibrationin the carrier frequency).

C. Influence of Unequally Spaced Fringes

The error of unequally spaced fringes in the SCPSmethod corresponds to a nonconstant phase step errorin the TPS method. This error can be observed when-ever the interferogram phase distribution differs from

a perfectly flat one. The amount of the error at everypoint of the calculated modulation map depends on thephase differences between N pixels used for calcula-tions. To illustrate the influence of this type of error onthe calculated modulation map, an additional qua-dratic phase term was added to the interferogramphase distribution. Figure 9 presents a cross sectionthrough the difference between error-free modulationand modulation distributions calculated using thethree algorithms considered. Modulation distributionwas assumed to be uniform over the whole interfero-gram. The phase shift between consecutive pixelsstarts at 90° and decreases to 80°.

The modulations calculated using algorithms 5Amodand 5Lmod are covered with parasitic fringes. The fre-quency of parasitic modulations is equal to the dou-bled frequency of two-beam interference fringes. Theamplitude of parasitic fringes is modulated by theerror envelope function as in the case of the carrierfrequency miscalibration. Algorithm 5Amod gives thelargest amplitude of parasitic fringes. This amplituderises with increasing carrier frequency miscalibra-tion. Algorithm 3Amod calculates modulation distri-bution without parasitic fringes. Generally, whenincreasing the miscalibration of the interferogramcarrier frequency, the difference between the error-free modulation and calculated modulation valuesincreases significantly. The error described abovecannot be removed by the normalization process as inthe case of spatial carrier miscalibration error. It can

Fig. 12. (Color online) Error in the modulation distribution calculated using three considered algorithms with the presence of a nonlinearrecording error and carrier frequency miscalibration. Constant modulation distribution has been assumed.

Fig. 13. Two-beam interferogram intensity modulation distributions for the static state of the AFM cantilevers calculated using threeconsidered algorithms. Applied carrier frequency was set to the value of four pixels per fringe.


be minimized, however, using the calculated curvespresented in Fig. 8. A detailed report on the compen-sation method will be the subject of a separate paper.

D. Influence of Nonlinear Recording

Of our interest is the case of the influence of a non-linear characteristic of a photodetector used on theevaluated modulation distribution using the SCPStechnique. Nonlinear recording results in the inten-sity distribution [1]:

I� � I � aI2 � bI3 � cI4 � . . . , (5)

where I � I�x, y� is a sinusoidal two-beam interfero-gram intensity distribution and a, b, c, . . . describenonlinearity coefficients. As the most common detec-tor nonlinearity is the second order one, we havechosen the coefficient a to be a variable, and all otherswere set to zero.

In Fig. 10 the differences between the error-freeand calculated modulation distributions for each con-sidered algorithm are presented. The nonlinearity ofa simulated interference signal is equal to �10% of amaximum intensity of a linear signal. The interfero-gram carrier frequency is set to cover four pixels byone fringe. Modulation distribution changes are de-

Fig. 14. First row shows two-beam interferogram intensity modulation distributions for the nonvibrating AFM cantilevers calculatedusing three considered algorithms. Applied carrier frequency was approximately six pixels per fringe. The second row shows the magnifiedupper left corners of the calculated modulation maps.

Fig. 15. Two-beam interferogram intensity modulation distributions for the first resonance mode of an AFM cantilever vibrating at21.1 kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.


scribed by the Bessel function J0 with the argumentin the range of 0–5��2 divided into 256 equidistantsamples. The interferogram fringes are assumed per-pendicular to the modulation change direction.

The modulation calculation results for three algo-rithms and the same numerical experimental condi-tions, but with parallel mutual orientation betweenthe carrier fringes and the modulation distributionchanges, are presented in Fig. 11. Horizontal crosssections through the calculated modulation distribu-tion reveal the error character.

In the case of the nonlinear recording error, onlythe algorithm 3Amod gives parasitic fringes at the fre-quency equal to the frequency of carrier fringes. Twoother algorithms reproduce the modulation distribu-tion quite smoothly. The evaluated modulation val-ues differ from the error-free ones depending on theamount of camera nonlinearity. As in the case ofcarrier frequency miscalibration the nonlinearity er-ror causes the scaling effect of the modulation distri-bution.

All above conclusions are preserved if suitable in-terferogram carrier frequency is used. Any miscali-bration of the interferogram carrier frequency leadsto parasitic fringes. The character of parasitic fringesis different for the algorithm used. The general con-clusion is that the frequency of this error is equal tothe frequency of carrier fringes. Figure 12 presentsthe cross sections through the difference betweenerror-free and evaluated modulation distributions

under the presence of nonlinear recording and thecarrier frequency miscalibration (60° phase shift be-tween the consecutive pixels).

4. Experimental Work

To check the applicability of the spatial carrier phaseshifting method for the modulation distribution de-termination several measurements were performed.The experimental setup was based on the Mach–Zehnder interferometer. The AFM cantilever withthe length of 300 m, width of 50 m, and thicknessof 1 m was used as the object under test. Its flatsurface allows applying an appropriate carrier fre-quency in the interferogram. The modulation distribu-tions for stationary and vibrating object at resonantfrequencies were determined using SCPS algorithmsintroduced in Section 2.

Modulation distributions calculated using threepresented algorithms for the case of a stationary ob-ject (quasi-uniform modulation distribution), and thecarrier fringes set perpendicular to the cantileverlength are presented in Fig. 13. The direction of thecarrier fringes was dictated by the measured objectdimensions. For carrier fringes parallel to cantileverlength only several fringes would appear on the objectsurface causing difficulties when analyzing resultsobtained. The carrier frequency was set to four pixelsfor the interferogram period.

The results of calculated modulation distributionsfor the experimental conditions similar to the ones

Fig. 16. Two-beam interferogram intensity modulation distributions for the second resonance mode of an AFM cantilever vibrating at142 kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.

Fig. 17. Two-beam interferogram intensity modulation distributions for the first torsional mode of an AFM cantilever vibrating at175 kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.


presented in Fig. 13, but with a carrier frequencymiscalibration are shown in Fig. 14. In this case theparasitic ripples at the double frequency with respectto the frequency of carrier fringes can be notedover the modulation map retrieved by algorithm5Amod. Two other algorithms evaluate modulation dis-tribution smoothly. A darker appearance of the mapevaluated by the 3Amod algorithm reveals its highersusceptibility to this kind of error (see Fig. 8).

Figures 15 and 16 present calculated modulationdistributions for the object vibrating at its first andsecond resonant frequencies (bending modes). Figure17 shows modulation distributions for the first tor-sional mode. In the case of sinusoidal vibrations themodulation distribution is described by the zero orderBessel function J0. The carrier frequency is set to fourpixels for one interference fringe, mutual perpendic-ular orientation between the modulation change di-rection and carrier fringes. High frequency errorcaused by the contrast envelope function changes (seeSubsection 3.A.1) is not observable in the calculatedmodulation maps.

Studies of the AFM cantilever at the static and vi-brating states show that the SCPS method can besuccessfully applied to the modulation distribution cal-culations. Strict measurement conditions such as propercarrier frequency and linear recording must be observed.

5. Conclusions

Single frame recording, simplicity of experimentalequipment and uncomplicated data processing arethe main advantages of the spatial carrier phaseshifting method, especially if adverse measurementconditions are met. In the paper the influence of themodulation distribution nonuniformity, spatial car-rier miscalibration, and nonlinear recording errors onthe evaluated intensity modulation of a two-beaminterferogram using the spatial carrier phase shiftingmethod has been studied. Modified three point andtwo five point algorithms were investigated.

The modulation distribution nonuniformity causedby the contrast envelope function or the interfero-gram bias changes leads to parasitic modulationsover the whole retrieved modulation map. Contrast

envelope function changes produce parasitic modula-tions with doubled spatial frequency with respect tothe interferogram carrier frequency. In this case thebest reproduction of the modulation distribution isprovided by algorithm 5Amod. To avoid this error theparallel mutual orientation between the contrast en-velope function and interferogram carrier fringesshould be provided. Parasitic fringes with the spatialfrequency equal to the interferogram carrier fre-quency appear if the interferogram bias changes areencountered. Algorithm 3Amod shows the best resis-tance to this kind of error.

Linear and nonlinear spatial carrier miscalibra-tions are the most common errors in the spatial car-rier phase shifting (SCPS) method. The former oneappears if incorrect tilt between reference and objectwavefronts is introduced. In this case algorithms3Amod and 5Lmod reproduce the modulation distribu-tion smoothly with the scaling effect; whereas thealgorithm 5Amod generates additional parasitic mod-ulation fringes (at double spatial frequency with re-spect to the interferogram carrier frequency) in theevaluated modulation map. Nonlinear spatial carriermiscalibration is caused by a nonflat object surface.This type of error affects the calculated modulationdistribution significantly. It causes parasitic ripplesin the modulation distribution evaluated by fivepoint algorithms. These algorithms, however, pro-vide higher accuracy of the modulation distributionreproduction than the three point one.

In case of nonlinear recording, only algorithm 3Amodproduces parasitic modulation fringes at the fre-quency equal to the frequency of carrier fringes overthe calculated modulation map. Five point algo-rithms smoothly reproduce the modulation distri-bution with the scaling effect only provided thatappropriate spatial carrier frequency is ensured.Any departure from this assumption leads to para-sitic ripples with the frequency equal to the fre-quency of carrier fringes.

Conclusions of our studies of the influence of selectedexperimental errors on the evaluated interferogrammodulation distribution calculated using three differ-ent SCPS algorithms are gathered in Tables 1 and 2.

Table 1. Carrier Fringes Perpendicular to the Direction of Modulation Changes

Contrast EnvelopeFunction

Bias DistributionNonuniformity

Spatial CarrierMiscalibration

Unequally SpacedFringes

NonlinearRecording

Modified 3Amod � � � � ��5Amod �� 5Lmod � � � � �

Table 2. Carrier Fringes Parallel to the Direction of Modulation Changes

Contrast EnvelopeFunction

Bias DistributionNonuniformity

Spatial CarrierMiscalibration

Unequally SpacedFringes

NonlinearRecording

Modified 3Amod �� 5Amod �� 5Lmod ��


The meaning of used symbols is as follows: “��”means significant influence of the selected error;“ � ” means moderate influence of the selected error;“� ” means resistance to the selected error; “��”means high resistance to the selected error.

Experimental works aimed at the visualization ofresonant modes of silicon AFM cantilevers under var-ious experimental conditions. In the case of sinusoidalvibrations the interferogram modulation distributionis described by the zero order Bessel function. Exper-imental studies proved the numerical findings for se-lected errors showing the limitations and applicabilityof spatial carrier phase shifting to the interferogramintensity modulation determination.

This work was supported by KBN (Polish Commit-tee for Scientific Research) grant 4T10C�001�27 andby MNiSW (Ministry of Science and Higher Education)grant N505 009 31�1425.

This paper is a revision of a paper presented at theSPIE Optics and Photonics Conference, August 2006,San Diego, USA. The paper presented there appearsunrefereed in Proc. SPIE 6292.

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Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination