Approaches in generalized phase shifting interferometry

Optics and Lasers in Engineering 43 (2005) 475–490

Approaches in generalized phase shiftinginterferometry

Abhijit Patil*, Pramod Rastogi

Applied Computing and Mechanics Laboratory, Swiss Federal Institute of Technology, 1015 Lausanne,

Switzerland

Received 14 May 2004; accepted 17 May 2004

Abstract

A sample of phase-shifting algorithms suitable for accommodating arbitrary phase steps is

passed in review. Although not exhaustive in nature, the paper describes a wide range of

concepts which have been applied to generalized phase shifting interferometry. Linear phase

shift miscalibrations and nonlinear sensitivity of the piezo electric device are known to

introduce errors in phase measurement. The study reveals that of the various algorithms

proposed most are suitable for compensating only one of these two error sources. Application

of a direct search stochastic algorithm would appear to be a promising step towards

characterizing the nonlinear response of the PZT.

r 2004 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of phase shifting interferometry makes possible the measurement ofphase in interferograms [1,2]. Given the need for obtaining precise automatedmeasurements, there has been a flurry of activity directed toward the design ofrobust algorithms immune to error sources.

Over the years, several phase shifting techniques have been reported which,primarily function by changing the path length of the reference beam of theinterfering object and reference beams. These can be broadly classified into temporal(phase shifting) and spatial (frequency transform) techniques [2,3]. Phase shifting

ARTICLE IN PRESS

*Corresponding author.

E-mail address: [email protected] (A. Patil).

0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.optlaseng.2004.05.005

interferometry using temporal techniques functions by either acquiring interfero-grams while the phase is stepped with respect to time or by acquiring theinterferograms with predefined phase steps. These conventional techniques however,are employed with phase steps which are typically multiples of p=2 radians. Toovercome this limitation, a generalized data reduction technique has been proposedwhich adds the facility of using arbitrary phase steps [4,5].

In interferometry, phase steps are usually introduced by piezoelectric actuators(PZT). In practice, these actuators show a nonlinear response. Apart fromnonlinearity, there are several other sources of errors, such as miscalibration,mechanical vibration during PZT translation, detector nonlinearity, etc. which affectthe accuracy of phase measurement [6–11]. While several algorithms have beenproposed to minimize these errors in conventional phase shifting interferometry, theestimation of the exact phase step and consequently the interference phasedistribution in the presence of error sources remains the main source of difficultyin phase shifting interferometry using arbitrary phase steps [12].

The aim of this paper is to review contemporary algorithms for the extraction ofunknown phase steps. The paper is restricted to cover a selection of procedures ingeneralized phase shifting interferometry and which in recent years has assumedsignificance because of the flexibility it offers in terms of selecting arbitrary phasesteps. Although not exhaustive in nature, the paper describes a wide range ofconcepts which have been applied to generalized phase shifting interferometry. Thepaper summarizes applications of concepts such as Fourier transform [12], ellipticalcurve fitting [13–15], statistical [16,17], spatio-temporal [18], iterative [19], andoptimization [20] to phase shifting interferometry in the context of arbitrary phasesteps. Algorithms based on other or related concepts and not included or cited in thepaper in no way demerit their importance.

Section 2 discusses algorithms for the determination of unknown phase stepswhich can subsequently be applied for the determination of interference phase.The Section starts with a discussion on spatial technique proposed by Lai andYatagai [12] to extract reference phase steps by the generation of Fizeau fringes.The method suggested by Kinnstaetter et al. [13], and by Farrell and Player [14,15]shows the application of algebraic fitting to phase shifting. The seminal work byCarr!e [21] and its detailed analysis is also presented. The statistical approachadopted by Kadono et al. [16] and by Cai et al. [17] shows that reference phasesteps can be computed by considering the statistical properties of the speckleintensity. The computationally exhaustive algorithm by Chen et al. [18] shows thatphase can be reliably extracted. The algorithm based on iterative least squaresestimation proposed by Han and Kim [19] is discussed. The algorithms proposed byJ .uptner et al. [22], and Stoilov and Dragostinov [23] which provide the flexibility ofusing arbitrary phase steps are also presented. Section 3 presents a case study inwhich a stochastic approach is used to determine the value of the arbitrary phasestep. This algorithm also provides the possibility of identifying the nonlinearcharacteristics of PZT to the applied voltage. Section 4 shows the iterative leastsquares fit technique which can be applied once the exact value of the reference phasestep has been obtained.

ARTICLE IN PRESSA. Patil, P. Rastogi / Optics and Lasers in Engineering 43 (2005) 475–490476

2. Algorithms for unknown phase determination

2.1. Real time reference phase estimation using fast-fourier- transform method

Lai and Yatagai [12] have proposed an algorithm which combines to thegeneralized phase shifting interferometry the direct evaluation of reference phases inreal time. The technique involves generation of Fizeau fringes using additionaloptical setup and the reference phase is evaluated from the parallel Fizeau fringes inthe interference plane. For the generalized phase shifting to calculate the interferencephase distribution correctly, the reference phase steps have to be accurately known.The technique functions as follows:

The equation for two-beam interferometer can be written as

IK ðx; yÞ ¼ I0ðx; yÞf1þ gðx; yÞ cos½jðx; yÞ þ aK �g; ð1Þ

where I0ðx; yÞ is the mean intensity, gðx; yÞ is the fringe contrast, jðx; yÞ is the phasedistribution to be measured and aK is the relative reference phase of N data frames(K ¼ 0; 1;y:;N 1). The straight line Fizeau fringes that are generated have thefollowing intensity equation

IðxÞ ¼ I0

0ðxÞf1þ g0ðxÞ cos½2pfx þ a�g; ð2Þ

where f is the spatial frequency of the straight fringes and a is the initial phase of thereference beam which is changed by either applying voltage to the PZT or by varyingthe laser frequency. The initial reference phase a in Eq. (2) is closely related to aK inEq. (1) and only differs in the origin of the phase as they are created in the sameinterferometer. Hence, the knowledge of initial phase a in Eq. (2) helps indetermining aK : The phase step a is determined by obtaining the Fourier transformof Eq. (2). The spectrum obtained can be written as

IðuÞ ¼Z

N

N

IðxÞexp ð2piuxÞ dx

¼ aðuÞ þ cðu f Þexp ðiaÞ þ c�ðu þ f Þexp ðiaÞ; ð3Þ

where

aðuÞ ¼Z

N

N

aðxÞexp ð2piuxÞ dx; ð4Þ

cðuÞ ¼1

2

ZN

N

aðxÞgðx; yÞexp ð2piuxÞ dx: ð5Þ

Here, aðuÞ and cðuÞ represent the Fourier transform of the intensity envelope and firstorder intensity spectrum, respectively. To separate the first order from the zero orderspectrum, the spatial frequency of the straight line fringes is made high. By using theband pass filter the first order function can be written as

I 0ðuÞ ¼ cðu f ÞexpðiaÞ ð6Þ

ARTICLE IN PRESSA. Patil, P. Rastogi / Optics and Lasers in Engineering 43 (2005) 475–490 477

The logarithm of the filtered spectrum on frequency f is

log ½I 0ðf Þ� ¼ log½cð0Þ� þ ia ð7Þ

The phase of log½cð0Þ� in Eq. (7) will be constant if the average intensity aðxÞ andfringe contrast gðxÞ are constant in time. Thus, the phase a for a particular frequencyf can be determined from Eq. (7) using

a ¼ Imflog ½I 0ðf Þ�g constant ð8Þ

The reference phases are evaluated from the straight line fringes at the same timethe interfering data is read. The evaluated reference phases are used in the datareduction method described in Section 4.

The technique has an advantage of using arbitrary phase steps and does notrequire the calibration of the PZT. In addition, this method contributes significantlyto minimizing sources of errors that are associated to the use of PZT, such as, itsinherent hysteresis and nonlinearity, temperature drift, and surrounding distur-bances, by its ability to evaluate in real time and during the acquisition of dataframes. However, this technique requires additional optical setup for the generationof Fizeau fringes and as mentioned by Larkin [3], in reality is only suitable for fringeswhich are not straight and equally spaced.

One of the problems associated while using Fourier transform techniques has beenthe perfect isolation of background intensity I0ðx; yÞ in Eq. (1). The completeisolation is a problem because the discontinuities in the spatial fringe patterns causethe sidelobes to expand into the entire frequency domain; the finite size of the filter toisolate the sidelobes causes artefacts in the spatial domain; and non-monotonicity ofphase for closed fringes introduces sign ambiguities in phase estimation [3]. Larkin

suggested subtraction of inter frame intensity so that the dc term is removed. The 2DHilbert-Fourier demodulation algorithm is applied to estimate the analytic image.Lastly, the inter-frame phase shifts are estimated and data reduction technique isapplied [3].

2.2. Lissajous figure and elliptical fitting applied to phase shifting interferometry

Kinnstaetter et al. [13] made use of Lissajous figures for the determination of errorsarising from detector nonlinearity, laser drift, and miscalibration of PZT in phaseshifting interferometry. In this method, a Lissajous figure is plotted using phase stepas a parameter with intensities obtained from two points on the fringe field separatedapproximately in phase quadrature. In this method, two detector elements from thearray of CCD are selected by interactive software which are in phase quadrature.The photo voltage of detector 1 is displayed as the abscissa, and the photo voltage ofdetector 2 is displayed as the ordinate of a Cartesian coordinate system. The brightspot forms a Lissajous figure which assumes circles if the phase difference of thedetector is in quadrature and the mean intensity, visibility, and sensitivity of thedetecting elements at the two points on the CCD array are equal. The authors


showed that the Lissajous figure convey the following information.

1. Nonequidistant dots on the circumference of the Lissajous figure show that thephase steps are not equidistant (a fact attributed to nonlinearity in PZT and thehysteresis inherent in its characteristic).

2. Incorrect calibration of the PZT results in open gap or overlap on thecircumference of the figure.

3. Lissajous figure shows higher harmonics if the average intensity of theinterference pattern varies during the data acquisition period.

4. Lissajous figure is also indicative of the mechanical vibration induced whilerecording of the interferogram. The low frequency vibrations move the brightspot towards the circumference while the high frequency moves the dot towardsthe centre of gravity of the figure.

5. The nonlinear characteristics of the detector array results in the degradation fromthe usual Lissajous geometries from ellipse or circle.

After identifying the characteristics of the Lissajous figure, an iterative algorithmis applied to adjust the physical step mechanism to obtain equal phase steps. Sincethe phase steps to the PZT are applied by changing the voltage, this iterativealgorithm is repeated several times until the Lissajous display shows that the phasedifference between the chosen pixels are in phase quadrature.

Farrell and Player [14,15] have suggested a method which utilizes a Lissajousfigure and ellipse fitting for the calculation of phase step between the acquired dataframes. This approach also computes the average intensity I0ðx; yÞ and the intensitymodulation in the interferograms. In this technique an ellipse is fitted by plotting onefringe profile of the interferogram against the other by the following method.

Eq. (1) can be written as

I1ðx; yÞ ¼ I01 þ r1 cos½jðx; yÞ þ a1�; ð9Þ

where I01 is the average intensity of the fringe bias, r1 is the amplitude of the intensitymodulation and a1 is the phase step . Similarly for the second frame and the phasestep of a2, Eq. (1) can be written as

I2ðx; yÞ ¼ I02 þ r2 cos½jðx; yÞ þ a2�: ð10Þ

Under the assumption a1 ¼ 0; the general equation for the ellipse becomes

I1 I01

r21

2 cos a2ðI1 I01ÞðI2 I02Þr1r2

þI2 I02

r22¼ sin2a2: ð11Þ

Subsequently, a conic is fitted to the points, produced by at least one period of thefringe profile, using the Bookstein method [24]. Eq. (11) can be written in the generalconic form as

aI21 þ bI1I2 þ cI22 þ dI1 þ eI2 þ f ¼ 0; ð12Þ

where the coefficients can be algebraically fitted.


The estimation of coefficients a, b, c, d, and e results in the determination of phasestep a2; I01; I02; r1 and r2: Once these values are known the interference phase jðx; yÞcan be calculated. Since this technique uses Bookstein method to fit a conic, thescattered and noisy data results in fitting a hyperbole instead of an ellipse. Given thatBookstein method is a forced fit method, several other ellipse specific methods whichare also computationally efficient and fit the scattered data perfectly to ellipse havebeen reported [25].

2.3. The Carr!e algorithm

Incidentally, the first algorithm proposed by Carr!e [21] used arbitrary phase stepsa. The advantage of Carr!e’s algorithm is that the phase shifter does not need to becalibrated. The algorithm can work with diverging as well as converging referencebeams. For the phase step a between the consecutive frames, the four intensityequations can be written as

I1ðx; yÞ ¼ I0ðx; yÞf1þ gðx; yÞcos ½jðx; yÞ 3a=2�g; ð13Þ

I2ðx; yÞ ¼ I0ðx; yÞf1þ gðx; yÞcos ½jðx; yÞ a=2�g; ð14Þ

I3ðx; yÞ ¼ I0ðx; yÞf1þ gðx; yÞcos ½jðx; yÞ þ a=2�g; ð15Þ

I4ðx; yÞ ¼ I0ðx; yÞf1þ gðx; yÞcos ½jðx; yÞ þ 3a=2�g: ð16Þ

The phase step aðx; yÞ is retrieved as follows:

aðx; yÞ ¼ 2 tan1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðI2 I3Þ ðI1 I4ÞðI2 I3Þ þ ðI1 I4Þ

sð17Þ

and the phase is obtained as

jðx; yÞ ¼ tan1 tan ða=2ÞðI1 I4Þ þ ðI2 I3ÞðI2 þ I3Þ ðI1 þ I4Þ

� �� ð18Þ

The algorithm is based on the assumption that only linear errors exist in PZTs, acase which is highly unlikely. Carr!e claimed that 110� is the best phase step withoutciting sufficient explanation. A detailed theoretical study by Kemeo et al. [26] usingTaylor expansion of the interference phase function has revealed that 110� is the bestphase step only for random intensity error, and not for phase shift or systematicintensity errors. The study has also revealed that phase step should be less than 120�

in order to minimize the second-order phase shift error. Furthermore, phase stepbetween 90� and 120� was considered to minimize second and third order intensityerrors. For the Carr!e’s algorithm to work properly the numerator in Eq. (18) must bepositive. However, this condition is true for perfect images only and may not hold inthe presence of error sources [26].


2.4. Phase step estimation based on speckle phase statistics

Kadono and Toyooka [16] propose a technique based on the statistics of a fullydeveloped speckle field. In optical interferometry the generation of speckles isconsidered as a noise and several researchers have made efforts to reduce thespeckles so as to increase the reliability of measurements. The random nature of thespeckle field is known to play the role of a standard phase object in a statistical sense.This phase though random in nature is regarded as a reference field same as adeterministic reference field in conventional interferometry. The method proposedby the authors, however, requires two diffusers in front of the testing object toproduce a fully developed speckle field. Another approach suggested by Cai et al.[17] is based on the same principle and does not require a diffuser to produce specklefields.

Although simulation results have been shown to extract arbitrary phase stepsefficiently, the practical feasibility of this statistical method would most likely dependon the sampling of the speckle intensity. The spatial resolution of CCD cameradictates the sampling number of the speckle intensity.

2.5. Max–min algorithm for computation of interference phase distribution

The algorithm proposed by Chen et al. [18] allows the use of uncalibrated phaseshifts for the estimation of phase distribution, and is insensitive to the spatialintensity variation across the interferograms. The spatial and temporal algorithmuses a maximum and minimum (max–min) technique that spans the number ofacquired interferograms. First, a comparison of each pixel in an interferogram withthe pixels at the corresponding location of acquired interferograms helps inremoving its sensitivity to spatial intensity variations. This is followed by the spatialcomparison of each pixel’s intensity with the intensities of two reference pixels withinthe same interferogram. The temporal variation of these spatial comparisons acrossall the acquired frames enables for estimating the phase distribution of the objectbeam.

Since the measurement of phase is obtained by selecting two points on theinterferogram, the problem of ambiguity in the determination of the sign remains.The problem is addressed by extending the procedure to three points on theinterferogram. The algorithm requires fifteen or more frames to work adequately.Hence, the computational load is high. The algorithm based on the principle of two-point correlation is efficient as it needs a small number of pixel pairs for calibration.

2.6. Numerical methods for reference phase steps determination

Han and Kim [19] proposed a computational algorithm where the reference phasesteps are considered to be unknown and iterative least squares technique is employedto extract the phase steps. This algorithm is immune to nonlinear and random errorsarising due to piezoelectric device. The interference equation of the Kth frame can be


written as

IK ¼ I0 þ I0g cos jþ aKð Þ: ð19Þ

Assuming that there are n pixels in an interferogram and K data frames areacquired, Eq. (19) for each pixel location i can be written as

IiK ¼ I0i þ I0igicos ji þ aK

� ¼ I0i þ micosji cosaK misinjisinaK

¼ I0i þ UicosaK VisinaK ; ð20Þ

where, the modulation intensity mi ¼ I0igi; and Ui ¼ mi cosji:Eq. (20) contains three unknowns: I0i; Ui; and Vi; and for K interferograms with n

pixels each, the number of equations is n � K : The number of unknowns in theseequations is 3n þ K 1:Hence, for unique solutions the number of equations shouldbe greater than the unknowns.

nKX3n þ K 1 ð21Þ

Under the assumption that Eq. (21) is satisfied, the least square iterative techniqueis applied for the computation of I0i; Ui; and Vi by minimizing the error function

P ¼Xn

i¼1

XK

j¼1

Iij I 0ij

� �2

: ð22Þ

Subsequently, phase aK for each frame is calculated and applied to the well knownfive bucket algorithm [9,10] which is effective for linear errors of the reference phaseshifts. The proposed algorithm can estimate the phase step value accurately onlywhen the initial guess is as close as possible to the reference phase step, a conditionwhich is quite difficult to satisfy.

2.7. Alternative to Carr !e algorithm for reference phase step determination

J .uptner et al. [22] proposed an alternative algorithm to Carr !e in which first thephase steps at each pixel location in an interferogram is computed. This is done byusing Eqs. (13)–(16). The equation for the estimation of the unknown phase step isobtained as

a x; yð Þ ¼ cos1 I1 x; yð Þ I2 x; yð Þ þ I3 x; yð Þ I4 x; yð Þ2 I2 x; yð Þ I3 x; yð Þ½ �

: ð23Þ

It is assumed that a x; yð Þ is constant for all points on the interferogram. InEq. (23) the outliers occur when the denominator is zero or close to zero and thesepoints are discarded in the calculation of average phase step aAVG: The averagephase step is used for the estimation of interference phase distribution using either

j x; yð Þ ¼ tan1 I3 I2 þ I1 I3ð ÞcosaAVG þ I2 I1ð Þcos2aAVG

I1 I3ð ÞsinaAVG þ I2 I1ð Þsin2aAVGþ

3aAVG

2; ð24Þ


or

j x; yð Þ ¼ tan1 I4 I3 þ I2 I4ð ÞcosaAVG þ I3 I2ð Þcos2aAVG

I2 I4ð ÞsinaAVG þ I3 I2ð Þsin2aAVGþ

aAVG

2: ð25Þ

The experimental results presented in [27] show that the average phase step isestimated from Eq. (23), with the value of the reference phase step a x; yð Þ varyingfrom 0 to p radians. After discarding the outliers, the value of the evaluated phasestep is found to lack in precision and may decrease from its optimum value for anoisy interferogram. The authors have also extended this approach to five frames.Similar work is reported by Stoilov and Dragostinov [23] for the computation ofarbitrary but constant unknown phase steps. These algorithms are suitable tominimize only linear calibration errors.

2.8. Stochastic algorithm applied to Optical Interferometry

Patil et al. [28] have recently proposed an application of a global search techniquecalled Probabilistic Global Search Lausanne (PGSL) to determine the arbitraryunknown phase steps. PGSL computes the global minimum of the least squares errorobjective function defined for the interference equation. PGSL has been shown toidentify the linear and nonlinear response of the PZT to the applied voltage. Thetechnique carries out one optimization at the same pixel location of the N acquireddata frames. About 100 optimizations/30 s are carried out and the errors due to PZTnonlinearity are computed. The true phase step is next estimated, averaged over theselected data points, and subsequently applied in the linear regression technique forthe estimation of interference phase.

A simple mathematical representation of the recorded interference fringe intensityat pixel ðx; yÞ of the N data frame is given by Eq. (1). Normally three data frameswould be sufficient for the extraction of the interference phase. However, thepresence of error sources can be minimized by acquiring additional data frames. Forthe N step phase shifting algorithm, the true phase shift a0K x; yð Þ due to nonlinearcharacteristics of PZT can be represented as [11]

a0K x; yð Þ ¼ aK x; yð Þ þ e1aK x; yð Þ þ e2aK x; yð Þð Þ2

2pþ e3

aK x; yð Þð Þ3

4p2; ð26Þ

where a0K x; yð Þ is the third order polynomial representation of the phase shift and e1is the linear and e2 and e3 represent nonlinear errors in phase shift. The mechanicalvibration experienced by the PZT during its translation can be representedmathematically by [7]

a00K x; yð Þ ¼ a0K x; yð Þ þ2pl

a sinKpfvib

fCCD

�ð27Þ

where, a is the amplitude of vibration and fvib and fCCD are the frequencies ofvibration and data acquisition, respectively. The interference fringes represented by


Eq. (1) can thus be written as

IK x; yð Þ ¼ I0 x; yð Þ 1þ g x; yð Þcos j x; yð Þ þ a00K x; yð Þ� ��

K ¼ 0; 1; 2; y;N 1: ð28Þ

Assuming that PZT has nonlinear characteristics defined by Eq. (27), PGSL isemployed to determine the exact phase step a00K x; yð Þ by computing the globalminimum of the least squares error objective function P x; yð Þ defined by

P x; yð Þ ¼XN1

K¼0

IK x; yð Þ I0 x; yð Þ 1þ g x; yð Þcos j x; yð Þ þ a00K x; yð Þ� �� 2

: ð29Þ

PGSL is a direct search technique that employs random sampling using aprobability distribution function (PDF) in order to locate the global minimum of auser defined objective function. PGSL functions by assuming uniform PDF over theentire search space and which is updated dynamically as search progresses such thatmore intensive sampling is performed in regions where good solutions are found.

The PGSL algorithm carefully samples the search space through four nested cyclescalled Sampling cycle, Probability updating cycle, Focusing cycle, and Subdomaincycle. Each cycle serves a different purpose in the search for the global optimum. Thesampling cycle permits a more uniform search over the entire search space. Searchspace is the set of all possible solution points. It is an N-dimensional space with anaxis corresponding to each variable. Probability updating and focusing cycles refinesearch in the neighbourhood of good solutions. Convergence to the optimumsolution is achieved by means of the subdomain cycle. The user initially defines theminimum and maximum values for each variable in the objective function. The exactphase step a00K x; yð Þ is computed at the end of the subdomain cycle. The function ofeach cycle is described in ref. [20].

3. A Case Study: an example of application of PGSL

Section 2 has described algorithms based on the premise of using arbitrary phasestep. As a case study, we have chosen PGSL to demonstrate its ability to estimate thearbitrary phase shift imparted by the PZT in the presence of mechanical vibration.The algorithm is initially tested by applying it to computer generated interferencefringes. Ten data frames are generated with step value a as 0.872664 radians (50�). Itis assumed that phase shift is constant over the entire interferogram. The bounds forthe variable in Eq. (29) are defined as follows:

0pI0pImax Imin

2; 0pgp1; 0pjp2p; and 0papp; ð30Þ

where Imax and Imin are the maximum and minimum values of the intensity at anypoint ðx; yÞ on the interference image. For 8 bit data Imax ¼ 255 and Imin ¼ 0: Thefringe visibility g is generated randomly between 0 and 1. The values for NS (numberof sampling cycles) and NPUC (number of probability updating cycles) are taken as2 and 1, respectively from observations for smooth objective functions while NFC


(number of focusing cycles) is selected to be 60V, where, V is the number of variablesin the function. The value for NSDC (number of subdomain cycles) which controlsthe precision is chosen to be 60.

The algorithm is applied to 1000 randomly selected pixel locations to calculatereference phase step by using the objective function defined in Eq. (29). Under theassumption that only first order error (suppose, e1 ¼ 5%) exists in PZT, six dataframes are sufficient for the exact estimation of the phase step. In this case the stepvalue is estimated to be 50.0125� with a relative error of 0.025% The step value canbe estimated with higher precision if more frames are acquired. In case of nine dataframes the error in computation of phase step is 0.006%. However, a trade-off ismade between the number of frames, computational time required, and the accuracydesired. Acquisition of a large number of frames makes the interference phasemeasurement prone to air turbulence, temperature fluctuations, and laser drift. Table1 shows the average step value obtained in function of the number of frames. For thesecond simulation we assume the vibration frequency fvib to be 10Hz for two

ARTICLE IN PRESS

Table 1

Number of frames and corresponding phase values for a=50� (assuming no error in calibration

Frames Step value

5 49.9905

6 50.0125

7 50.0098

8 50.0129

9 50.003

52.45

52.5

52.55

52.6

52.65

52.7

52.75

52.8

52.85

52.9

6 8 10 12 14

Number of Frames

Ste

p V

alu

e (D

egre

es)

a = 0.5

a = 1.0

Fig. 1. Plot of phase step versus number of frames for two different amplitudes of vibration, a ¼ 0:5 and

1.0.

A. Patil, P. Rastogi / Optics and Lasers in Engineering 43 (2005) 475–490 485

amplitudes of vibrations (a ¼ 0:5; 1). The effect of the amplitude of vibration on theestimation of the exact phase step a; in the presence of first order calibration error(e1 ¼ 5%), is shown in Fig. 1. The step value is determined by averaging over thirteenand fourteen data frames. Figs. 2a and b show the amplitude of vibration obtained,for a ¼ 0:5 and 1.0, respectively. Fig. 3 shows that the frequency is estimatedaccurately for various amplitudes of vibration.

ARTICLE IN PRESS

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

6

(a)

7 8 9 10 11 12 13 14

0.937

0.938

0.939

0.94

0.941

0.942

0.943

0.944

0.945

0.946

0.947

0.948

6 7 8 9 10 11 12 13 14

Number of Frames

Number of Frames

Am

plit

ud

eA

mp

litu

de

(b)

Fig. 2. (a) Plot of the amplitude of vibration versus number of frames for a ¼ 0:5: (b) Plot of the

amplitude of vibration versus the number of frames for a ¼ 1:0:

A. Patil, P. Rastogi / Optics and Lasers in Engineering 43 (2005) 475–490486

4. Linear Regression Technique for the Estimation of Phase

Once the exact phase shift is identified the subsequent step is to apply linearregression (least squares method) outlined in Refs. [4,5] for the determination ofinterference phase distribution. The equation for interference can be written as

IK x; yð Þ ¼XN1

K¼0

I0 x; yð Þ þ _ x; yð Þcosa00

KAVG þ | x; yð Þsina00

KAVG

h ið31Þ

where _ x; yð Þ ¼ I0 x; yð Þg x; yð Þcos j x; yð Þ½ �; | x; yð Þ ¼ I0 x; yð Þg x; yð Þsin j x; yð Þ½ �;and a

00

KAVG is the average step value obtained by optimizing least squares errorobjective function over selected data frames. The interference phase distribution canbe determined using

j x; yð Þ ¼ tan1| x; yð Þ_ x; yð Þ

: ð32Þ

Since Eq. (31) is linear with respect to unknown coefficients I0 x; yð Þ; _ x; yð Þ; and| x; yð Þ; we can use least squares fit technique to minimize the equation. The errorfunction P x; yð Þ can thus be written as

P x; yð Þ ¼XN1

K¼0

I0 x; yð Þ þ _ x; yð Þcoska00

AVG

h

þ| x; yð Þsinka00

AVG IK x; yð Þi2: ð33Þ

ARTICLE IN PRESS

9.95

10

10.05

10.1

10.15

10.2

10.25

10.3

5 6 7 8 9

Number of frames

10 11 12 13 14

Fre

qu

ency

(Hz)

a = 0.5

a = 1.0

Fig. 3. Plot of vibration frequency versus number of frames for two different amplitudes of vibration,

a ¼ 0:5 and 1.0.

A. Patil, P. Rastogi / Optics and Lasers in Engineering 43 (2005) 475–490 487

For the best fit the linear regression function in Eq. (33) should be minimized(minPðx; yÞ: This is done by obtaining the first derivative of the residual errorfunction with respect to the unknown coefficients I0 x; yð Þ; _ x; yð Þ and | x; yð Þ; andequaling them to zero:

@P x; yð Þ@I0 x; yð Þ

¼XN1

K¼0

I0 x; yð Þ þ _ x; yð ÞcosKa00

AVG

h

þ| x; yð ÞsinKa00

AVG IK x; yð Þi2¼ 0; ð34Þ

@P x; yð Þ@_ x; yð Þ

¼XN1

K¼0


AVG

h

þ | x; yð ÞsinKa00

AVG IK x; yð Þi2¼ 0; ð35Þ

@P x; yð Þ@| x; yð Þ

¼XN1

K¼0


AVG

h

þ| x; yð ÞsinKa00

AVG IK x; yð Þi2¼ 0: ð36Þ

Eqs. (34–36) resulting from first order derivatives can be written in matrix form as

NPN1

K¼0

cosa00KAVG

PN1

K¼0

sina00KAVG

PN1

K¼0

cosa00KAVG

PN1

K¼0

cos2a00KAVG

PN1

K¼0

sina00KAVGcosa00KAVG

PN1

K¼0

sina00KAVG

PN1

K¼0

sina00KAVGcosa00KAVG

PN1

K¼0

sin2a00KAVG

2666666664

3777777775

I0 x; yð Þ

_ x; yð Þ

|ðx; y

264

375

¼

PN1

K¼0

IK

PN1

K¼0

IKcosa00KAVG

PN1

K¼0

IKsina00KAVG

2666666664

3777777775

ð37Þ

Solution to the above matrix results in a wrapped phase map, which is modulo 2paccording to Eq. (32). Phase demodulation techniques developed over the years canbe used to remove the p phase ambiguities between the adjacent pixels in thewrapped phase map to get the interference phase j x; yð Þ:


5. Conclusion

To summarize, this paper has presented algorithms which consider reference phaseas an unknown parameter. The algorithm by Lai and Yatagai requires additionaloptical setup for the generation of Fizeau fringes. The technique of ellipse fittingapplied to interference intensity equations, proposed by Farrell and Player, usesBookstein algorithm which is a forced fit method and may fit the parametersobtained into hyperbole for the noisy data. The Carr!e algorithm is effective tominimize the first order linear calibration error but is not suited to minimize higherorder errors inherent in PZT. The algorithm also has restrictions on the referencephase steps to be used to minimize error sources. The statistical approach proposedby Cai et al. to extract reference phase steps will strongly depend on the spatialresolution of CCD camera which in turn will dictate the sampling number of thespeckle intensity. The max–min algorithm proposed by Chen et al. is computation-ally exhaustive and requires large number of data frames (15 or more) for reliableoperation. The algorithm based on iterative least squares estimation proposed byHan and Kim is suitable only for linear first order errors in PZT and requires theinitial guess of the phase shifter to be as close as possible. The concept of identifyingthe reference phase steps proposed by J .uptner et al. is suitable for only linearmiscalibration error and can produce large number of outliers for noisyinterferograms. Although the five frame algorithm proposed by Stoilov andDragostinov gives the flexibility of using arbitrary phase steps, its use is limitedonly to linear errors encountered in the PZT. The stochastic algorithm to extract thephase step in the presence of vibration appears to be a promising tool in metrology.Most of these algorithms are complemented well by the Gaussian least squares errorfit technique for accurate measurement of phase in interferometry applications.

Acknowledgements

This research is funded by the Swiss National Science Foundation.

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Approaches in generalized phase shifting interferometry

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