ARTICLE IN PRESS

0143-8166/$ - se

doi:10.1016/j.op

�CorrespondE-mail addr

pitarresi@dima

(B. Zuccarello).

Optics and Lasers in Engineering 45 (2007) 631–636

Limitation of carrier fringe methods in digital photoelasticity

A. Ajovalasit�, G. Pitarresi, B. Zuccarello

Dipartimento di Meccanica, Universita di Palermo, Italy

Available online 28 September 2006

Abstract

The carrier fringes method has been proposed in digital photoelasticity in combination with techniques such as Fourier transform and

phase shifting method, without considering the influence of the isoclinics on the isochromatic patterns analysis. Unlike other optical

methods as moire and holographic interferometry, in photoelasticity the light intensity emerging from a circular polariscope is related to

both the isochromatic retardation and the isoclinic parameter. As it is shown by the theoretical analysis, owing to the misalignment

between the principal stresses in the model and in the carrier, the computed retardation is affected by an error which is the same for all

photoelastic methods based on the use of carrier fringes. Consequently, the photoelastic analysis carried out by methods that use carrier

fringes cannot be applied as a full-field technique. In detail, numerical simulations show that the retardation error is comparable (less

than 0.05 fringe orders) with that of other photoelastic methods provided that the misalignment between the principal stresses in the

model and in the carrier is less than 301. On the contrary, in the model zones where the misalignment is higher than 301, the retardation

measurement can be affected by non negligible errors (up to 0.25 fringe orders).

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Experimental mechanics; Digital photoelasticity; Fourier transform; Phase shifting; Carrier fringes

1. Introduction

The enhancement and diffusion of the modern digitalimage processing systems have given a noticeable impulseto the automated digital photoelasticity [1–3]. In particular,various methods have been developed:

�

Phase shifting methods [4–9]. � Fourier transform techniques [10–12]. � Spectral content analysis (SCA) techniques [13]. � RGB photoelasticity [14–18]. � Other methods using white light [19–21].

Carrier fringe methods are widely employed in inter-ferometry [22]. The use of a carrier fringe systems as thatused in the Babinet method of compensation [23], has beenalso proposed in digital photoelasticity, in combinationwith the Fourier transform method [10] and the phase

e front matter r 2006 Elsevier Ltd. All rights reserved.

tlaseng.2006.08.008

ing author. Tel.: +390916657 135; fax: +39 091484334.

esses: ajovalasit@dima.unipa.it (A. Ajovalasit),

.unipa.it (G. Pitarresi), zuccarello@dima.unipa.it

shifting method [24]. The Fourier transform method,unlike phase stepping based techniques [24], benefits fromthe advantage of using a single image to perform theanalysis, lending itself also as a potential suitable techniquefor the analysis of time dependent fields.Unlike other optical methods, as moire and holographic

interferometry the effect of carrier fringes in photoelasticitydepends however on two parameters: the isochromaticretardation d and the orientation of the maximum principalstress aU Unfortunately, for both the Fourier transformmethod [10] and the phase shifting method with carrierfringes [24], the evaluation of the retardation has beencarried without considering the influence of the isoclinicparameter on the isochromatic pattern, i.e. by consideringthe principal stresses of the model always aligned withthose of the carrier; consequently, the use of carrier fringescan lead to high errors on the retardation measurement.The influence of the misalignment between the principal

stresses in the carrier and in the model has been studied inRef. [11] as far as concerns the Fourier transform method.This paper considers in general the use of carrier fringes inphotoelasticity; its main aim is to highlight the limita-tions related to the use of a carrier fringe system in

ARTICLE IN PRESSA. Ajovalasit et al. / Optics and Lasers in Engineering 45 (2007) 631–636632

photoelasticity, caused by the misalignment between theprincipal stresses in the carrier and in the model.

2. General theory

In photoelastic methods with carrier fringes a carriersystem of equidistant straight fringes is superimposed tothe stressed model placed into a circular polariscope(Fig. 1a). Such a carrier system of fringes can be obtainedby using a quartz wedge or, as in Ref. [11], by means of asimple photoelastic specimen subjected to bending. Thelight intensity emerging from such a polariscope isprovided by the following formula given by the well knownJones matrix calculus [25]:

I ¼ I0 � I1½sin2ða� acÞ cos 2pðdc � dÞ

þ cos2ða� acÞ cos 2pðdc þ dÞ� þ I2. ð1Þ

In Eq. (1) the upper (+) and the lower (�) signs are validfor light field and dark field respectively, I0 is thebackground intensity, I1 is the term related to theisochromatic fringes, I2 is the noise, ac and a are the anglesdefining the orientation of the maximum principal stress inthe carrier and in the model respectively (Fig. 1b), dc is theretardation in the carrier, d is the unknown retardation inthe model that for a plane stress field is related to theprincipal stress difference s1�s2 by the following relation-ship:

d ¼Cd

lðs1 � s2Þ, (2)

where C is the stress-optic coefficient, d is the thickness ofthe model and l is the wavelength of the light source used.

polarizer

carrier

light source

TVC

λ/4 plate

λ/4 plate

PC

model

analyzer

(a)

Fig. 1. Dark field circular polariscope: (a) setup used in the experimental tests

and (b) orientation of the optical elements (P,A ¼ polarizers, Rp, Ra ¼ quarte

stress in the carrier, a, ac ¼ orientation of the maximum principal stresses in t

From Eq. (1) it is possible to observe how inphotoelasticity, unlike others optical methods as moireand holographic interferometry, the light intensity isrelated to the two parameter describing the stress field,i.e. the retardation d and the angle a. Only in particularconditions (a� ac ¼ 01 or 7901), the light intensitydepends only on the retardation. As an example fora� ac ¼ 01 Eq. (1) becomes

I ¼ I0 � I1 cos 2pðdc þ dÞ þ I2. (3)

Since the retardation dc is known, Eq. (3) contains fourunknowns I0, I1, I2 and d, where the last of them (d) is theinteresting one.The above reported basic relationships are valid for all

the optical methods that use a carrier fringes system; inparticular they are valid for both the Fourier transformmethod and the phase shifting method with carrier fringes.

3. Review of the Fourier transform method with carrier

fringes

In the presence of a proper carrier fringe system thefrequencies of the terms I0 and I2 are, respectively, lowerand higher than the carrier one; consequently such termscan be removed by means of a proper filtering generallyperformed by using the Fourier transform. This providesthe so called inphase signal If and inquadrature signal Iq; ifthe principal stresses in the model and in the carrier arealigned (a� ac ¼ 01 or a� ac ¼ �901), such signals arerelated to the retardations by the following relationships:

I f ¼ �I1 cos 2pdt, (4)

Iq ¼ �I1 sin 2pdt (5)

P

A

RpRa

�2c

�2

�1

�1c

�c

�

(b)

(TVC ¼ CCD camera, PC ¼ personal computer with digital frame board)

-wave plates, s1, s2 ¼ principal stresses in the model, s1c, s2c ¼ principal

he model and in the carrier).

ARTICLE IN PRESSA. Ajovalasit et al. / Optics and Lasers in Engineering 45 (2007) 631–636 633

with

dt ¼ dc � d, (6)

where in Eq. (6) the upper (+) and the lower (�) signs referto the conditions a�ac ¼ 01 and a�ac ¼7901, respec-tively.

From Eqs. (4)–(5) it follows:

tan 2pdt ¼sin 2pdtcos 2pdt

¼�Iq

�I f, (7)

where in Eq. (7) the upper (+) and the lower (�) signs referto light field and dark field circular polariscope respec-tively. Since the carrier retardation dc is known, Eq. (7),taking account of Eq. (6) allows the user to evaluate theunknown model retardation d.

3.1. Influence of the isoclinic parameter

In general the conditions a� ac ¼ 01 or 7901 are notsatisfied at all the model points since a varies from point topoint. Setting:

ja� acj ¼ ar ð0�parp90�Þ, (8)

the analysis of the influence of the isoclinic parameter onthe retardation d, leads to the following error d0�d [11]:

d0 � d ¼1

2ptan�1ðj cos 2arjtg2pdÞ � d. (9)

The analysis of Eq. (9) shows the following feature ofthe error d0�d: for ar ¼ 451 and d ¼ 0:25þ k=2(k ¼ 0� 1� 2� . . .) the retardation error jumps from�0.25 to 0.25, i.e the maximum error is equal to 0.25fringe orders. From these results, it is possible to state thatthe isoclinic parameter influences significantly the evalua-tion of the retardation in the zones of model where theprincipal stresses cross bisect the principal directions crossof the carrier (ar ¼ 451) and the retardation is around0.25+k/2 (k ¼ 0� 1� 2� . . .). Therefore, the methodcannot be used for a full field retardation analysis.Obviously, the retardation error can be restrained byexcluding a proper angle range centered around ar ¼ 451,but this need the prior knowledge of the isoclinicparameter.

Fig. 2. Experimental isochromatic fringes in dark field: model in pure bending a

451 (c) to the model axis.

As an example, Fig. 2 shows the experimental dark fieldisochromatic fringe pattern of the model in pure bendingalone (a) and of the model with the carrier whose axis isclockwise rotated respect to the model axis of 01 (b) and451 (c). Fig. 2c shows clearly the discontinuities (bifurca-tion of the fringes) of the isochromatics that occur at thezone of the model where d ¼ 0:25þ k=2.

4. The use of carrier fringes with the phase shifting method

Unlike Fourier transform, in the phase shifting method[24] the inquadrature signal is generated by opportunephase shifting of the carrier fringes system. The formula-tion of the method as outlined in [24] is based on theassumption that the principal stresses in the model arealigned with those of the carrier. By referring to the schemein Fig. 1, the method is based on four acquisitions of theisochromatics in a circular polariscope both in light anddark field:

(A)

lone

Carrier fringes in a generic initial position:� 1st acquisition: isochromatics in light field, intensity

Ia,� 2nd acquisition: isochromatics in dark field, inten-

sity Ib.

(a),

(B)

Carrier fringes shifted such that a 901 phase variationis introduced:� 3rd acquisition: isochromatics in light field, inten-

sity Ic,� 4th acquisition: isochromatics in dark field, inten-

sity Id.

Since it is assumed that a� ac ¼ 01, Eq. (3) applies. Byassuming also that I0 and I2 are the same for each set ofacquisitions (a, b) and (c, d), and that I1 is the same in allfour acquisitions, the following relations are derived:

Ia ¼ I0 þ I1 cos 2pðdc þ dÞ þ I2, (10)

Ib ¼ I0 � I1 cos 2pðdc þ dÞ þ I2, (11)

I c ¼ I0 þ I1 cos 2pðd0c þ dÞ þ I2, (12)

model with carrier whose axis is at a clockwise angle of 01 (b) and

ARTICLE IN PRESSA. Ajovalasit et al. / Optics and Lasers in Engineering 45 (2007) 631–636634

Id ¼ I0 � I1 cos 2pðd0c þ dÞ þ I2. (13)

Since the carrier fringes system is shifted such that

2pd0c ¼ 2pdc þp2, (14)

Eqs. (12) and (13) can also be written as

I c ¼ I0 � I1 sin 2pðdc þ dÞ þ I2, (15)

Id ¼ I0 þ I1 sin 2pðdc þ dÞ þ I2, (16)

Subtracting Eqs. (11) and (15) respectively, from Eqs.(10) and (16) now provides

Ia � Ib ¼ 2I1 cos 2pðdc þ dÞ, (17)

Id � I c ¼ 2I1 sin 2pðdc þ dÞ. (18)

From Eqs. (17)–(18) it follows that

tan 2pðdc þ dÞ ¼sin 2pðdc þ dÞcos 2pðdc þ dÞ

¼Id � I c

Ia � Ib, (19)

where such relationship, formally identical to that on validfor the Fourier transform method [11], allows for thedetermination of the unknown retardation d when theretardation dc of the carrier fringes is known. In the casethat a� ac ¼ �901, from Eq. (1) and with analogousprocedure it follows:

tan 2pðdc � dÞ ¼sin 2pðdc � dÞcos 2pðdc � dÞ

¼Id � I c

Ia � Ib. (20)

4.1. Influence of isoclinics

As already evidenced with the Fourier transformmethod, the assumption a� ac ¼ 01 or 7901 is not validin general since the isoclinic parameter changes throughoutthe model in an unpredictable way. In order to account forsuch variability it is necessary to refer to the general Eq. (1)

Fig. 3. Retardation error d0�d versus ar for d

and then proceed in a similar sort of way as done with theFourier transform method. By considering for instance01oarp451, and referring to Eqs. (1) and (14), therelationships (10), (11), (15) and (16), describing the lightintensities (Ia, Ib, Ic and Id) from the four acquisitions, nowbecomes:

Ia ¼ I0 þ I1½sin2ðarÞ cos 2pðdc � dÞ

þ cos2ðarÞ cos 2pðdc þ dÞ� þ I2, ð21Þ

Ib ¼ I0 � I1½sin2ðarÞ cos 2pðdc � dÞ

þ cos2ðarÞ cos 2pðdc þ dÞ� þ I2, ð22Þ

I c ¼ I0 � I1½sin2ðarÞ sin 2pðdc � dÞ

þ cos2ðarÞ sin 2pðdc þ dÞ� þ I2, ð23Þ

Id ¼ I0 þ I1½sin2ðarÞ sin 2pðdc � dÞ

þ cos2ðarÞ sin 2pðdc þ dÞ� þ I2. ð24Þ

Hence Eq. (19) now provides a retardation d0 differentfrom the actual retardation d:

tan 2pðdc þ d0Þ ¼Id � I c

Ia � Ib

¼sin2 ar sin 2pðdc � dÞ þ cos2 ar sin 2pðdc þ dÞ

sin2 ar cos 2pðdc � dÞ þ cos2 ar cos 2pðdc þ dÞ. ð25Þ

Eq. (25) coincides with that one which was derived forthe Fourier transform method [11]. Therefore the errorformula, given by Eq. (9) is valid also for the phase shiftingmethod with carrier fringes. Consequently, it can beconcluded that the influence of the isoclinic parameter isidentical for both the methods based on a carrier fringessystem, i.e. Fourier transform and phase shifting methods.

ifferent values of the actual retardation dU

ARTICLE IN PRESSA. Ajovalasit et al. / Optics and Lasers in Engineering 45 (2007) 631–636 635

5. Attenuation of the error caused by the isoclinic parameter

It has been shown that photoelastic methods based onthe use of a carrier fringes systems can give rise to aretardation error, described by Eq. (9), whenever the modelprincipal stresses are not aligned with the carrier principalstresses.

Fig. 3 shows such an error as a function of themisalignment angle ar, for different values of the actualretardation d. It can be observed that, for any d, the error isalways higher than 0.05 fringe orders for values of arincluded in the range 4517151. Therefore, if the isoclinicparameter is known, the retardation computed from asingle acquistion of isochormatics will be acceptable if theuser is able to position the carrier fringes such to keep arout of the previously mentioned range throughout all theanalysed area.

For the typical case where the isoclinic parameter isunknown, the error in the retardation evaluation can bereduced by taking three different acquisitions, eachcorresponding to an appropriate orientation of the carrierfringes, in order to acquire three different values of theretardation for each point of the three images: d01, d

02, and

d03. These will be affected by the following error (absolutevalue) with respect to the actual retardation d:

ei ¼ jðd0i � dÞj; i ¼ 1; 2; 3. (26)

Fig. 4 refers to three acquisitions with carrier orienta-tions equally shifted of 301, and acquired in particular with

Fig. 4. (a) Orientation of the maximum principal stress in the carrier for each

retardation given by the three acquisitions for a generic position of the maximum

(c) average error (from Eq. (27)) for three different values of actual retardatio

ac ¼ 151, 451 and 751 (Fig. 4a). Fig. 4b shows the errors ofeach acquisition as a function of the angle a for an actualretardation of d ¼ 0:15. Fig. 4c shows the error resultingfrom considering the average retardation among d01, d

02, and

d03 i.e. assuming:

d0avg � d ¼d01 þ d02 þ d03

3

� �� d

¼1

3½ðd01 � dÞ þ ðd02 � dÞ þ ðd03 � dÞ�. ð27Þ

Some comments arises in particular from the analysis ofFig. 4:

�

of

p

n.

for values of the actual retardation smaller than 0.15fringe orders the error have a soft fluctuation and theresidual error on the average of the three measuredretardations is smaller than 0.055 fringe orders;

� when the actual retardation is higher than 0.15 fringe

orders the error presents a marked peak only when themaximum principal stress in the model is oriented ataE01, �301, �601, �901; i.e. when one of the threeacquired images have ar ¼ 451.

� the maximum error (occuring at d ¼ 0:25) is reduced

from 0.25 fringe orders (error with a single acquisition)to about 0.08 fringe orders (when performing multipleacquisitions and computing the average value asdescribed above).

the three acquisitons (ac ¼ 151, ac ¼ 451 ed ac ¼ 751); (b) error on the

rincipal stress in the model, a, and for an actual retardation of d ¼ 0:15;

ARTICLE IN PRESSA. Ajovalasit et al. / Optics and Lasers in Engineering 45 (2007) 631–636636

All the previous considerations have been based on valuesof the actual retardation included in the range 0pdp0.25.

Their extension to higher values of d is though obvious.

This simple analysis shows that the error can berestrained to relatively low values at the cost of anincreased number of acquisitions. In doing so one of themost peculiar advantages of the Fourier transform method,i.e. the ability to perform a complete photoelastic analysiswith just one experimental acquisition, is seriously affectedby the need to perform more acquisitions. This is evenworst the case with the phase shifting method which willtrebled the number of its already conspicuous number ofacquisitions, going from four to twelve if adopting theexample presented in this work.

6. Conclusions

This work presents some general limitations concerningthe use of a carrier fringes system in photoelasticity.

The theoretical analysis shows that, because of themisalignment between the principal stresses in the modeland in the carrier, the isochromatic retardation computedby any photoelastic method employing carrier fringes, canbe affected by a significant error. In particular, such errorincreases with the misalignment and reaches its maximumof 0.25 fringe orders at the points of the model where theprincipal stresses bisect the carrier principal stresses andthe actual retardation is equal to 0.25+k/2(k ¼ 0� 1� 2� . . ..). Thus the methods employing carrierfringes are not suitable for full field isochromaticsphotoelastic analysis.

However, in principle carrier fringes methods can beused in limited area of the examined model where theprincipal stress cross misalignment is restricted and theretardation error can be tolerated. As an example formisalignment angles outside the range of 4517151 the erroris smaller than 0.05 orders. Hence if the isoclinic parameterdistribution of the model is known, methods employingcarrier fringes might be implemented for the analysis oflimited model zones, by taking care of orienting the carrierfringes as above mentioned.

In turn, the retardation error can be reduced over thewhole range of principal stress angles, by employingmultiple acquisitions of the isochromatic patterns obtainedby rotating properly the carrier fringes. Unfortunately, thiscan seriously affect the simplicity and versatility ofmethods based on the use of a carrier fringe system, asFourier transform and phase shifting method. In particularthe practical convenience of accomplishing the analysiswith the acquisition of a single image, typical of the Fouriertransform method, is lost.

Acknowledgements

The research work described in this paper has beenfinanced through research grants (60%) allocated by theUniversity of Palermo.

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