New method for optical object derotation

Carlos P�eerez L�oopeza, Fernando Mendoza Santoyoa,*, Mois�ees Cywiaka,Bernardino Barrientosa, Giancarlo Pedrinib

a Centro de Investigaciones en Optica, A.C., Loma del Bosque 115, Le�oon, Gto., Mexico 37150b Institut f€uur Technische Optik, Universit€aat Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany

Received 15 August 2001; received in revised form 16 January 2002; accepted 1 February 2002

Abstract

A new optical method capable of measuring out-of-plane deformations of rotating objects based on the Fourier

transform phase decoding technique is presented. Digital holography is used to test the method, which digitally der-

otates one of the holograms at its phase reconstruction stage to accurately remove object rotation fringes, uniquely

rendering phase maps that quantitatively show the out-of-plane deformation. Commonly, object derotators are based

on creating standing images of the rotating object under study. Typically, this is achieved by means of a rotating prism

that has to be precisely synchronised with the object rotation. In contrast, this new method eliminates the need of using

the expensive mechanical servomechanisms contained in the commercially available optomechanical derotators by

using double pulsed digital holography in conjunction with Fourier optics. � 2002 Published by Elsevier Science

B.V.

Keywords: Optical derotator; Digital holography

1. Introduction

Fringes from a rotating object relate mainly to,(a) deformations due solely to its rotation, forinstance, stresses due to the centrifugal force andout-of-plane vibrations, among others, and (b)the object angular displacement. Thus, a charac-teristic of these fringes is that due to the objectangular displacement a phase term is added to thefringes resulting from the deformation only. In

order to eliminate correctly the object angulardisplacement two experimental variables need tobe considered further. One is speckle decorrela-tion, present if the object angular displacement islarger than the average speckle diameter on theCCD sensor. The other, intrinsically connected tothe former, is present when the acquisition of thetwo digital holograms is made within a very shorttime lapse, such that the data due to the objectdeformation, between the two digital holograms,may be insufficient to measure a significantchange in it. Hence, when eliminating the con-tribution to the deformation only fringes from therotating object, three variables must be taken intoconsideration: (A) the additional phase term from

15 March 2002

Optics Communications 203 (2002) 249–253

www.elsevier.com/locate/optcom

*Corresponding author. Tel.: +52-4773-1017; fax: +52-4717-

5000.

E-mail address: fmendoza@foton.cio.mx (F. Mendoza San-

toyo).

0030-4018/02/$ - see front matter � 2002 Published by Elsevier Science B.V.

PII: S0030-4018 (02 )01169-0

the angular displacement, (B) the speckle decor-relation and (C) the time lapse between digitalhologram acquisition. The last two are experi-mentally controlled, while the first, also intrinsi-cally attached to the other two, may be calculatedfrom object rotation parameters and digitally in-troduced as a phase term into one of the digitalholograms. The latter is the subject of this paper.The phase due to object deformation must beindependent to that belonging to object rotation,otherwise the shape of the deformation fringeswill be distorted from the addition of a phasemodulating term belonging to the object rotation.Some solutions have been proposed [1–3] tocontrol variables from points (B) and (C), e.g.,the use of pulsed lasers to freeze the object mo-tion between the acquisition of the two digitalholograms and the use of on-axis illuminationoptomechanical derotators [4–9] that eliminatethe object rotation fringes.Commercially available optomechanical dero-

tators rely on a roof-edged rotating prism that isturned at half the angle of the object. This isdone by a complex servomechanism that servesto synchronise the prism and object rotationcreating a standing image of the object understudy.An alternative and simpler method that does

not use the rotating prism and servomechanism isthe one proposed in this paper. It is based on thedigital manipulation of the phase as found in theFourier domain, using the underlying principles of

digital holography [10] to introduce a compensat-ing phase in one of the digital holograms, recov-ering exclusively the dynamic out-of-plane objectdeformation. It is then an optical method thatcompletely removes the object rotational motionwithout the need of the optomechanical derotator,thus allowing the sole analysis of the object de-formation while in rotation. Experimental resultsshow the object phase map due to, exclusively, itsdeformation.We would like to emphasise at this point that

the digital holograms are taken while the objectcontinuously rotates, in contrast to the conven-tional technique where the digital holograms aretaken before and after rotation.

2. Mathematical model

The mathematical model that applies to phasesubtraction of digital holograms will be brieflymentioned, since more detailed discussions can befound elsewhere [10]. Consider an out-of-planesensitive digital holography set up as shown inFig. 1. Assume that there exists a deformationbetween two specific object rotation positions, 1and 2, and that the object angular displacement issuch that variables in points (B) and (C) above arecontrolled. All variables are ðxc; ycÞ dependent,where the origin of this co-ordinate system is fixedat the center of the CCD sensor. Let rðxc; ycÞ andoðxc; ycÞ be the reference and object waves, re-

Fig. 1. Experimental set-up with its sensitivity vector coming out-of-plane from the rotating air fan. The object is rotating perpen-

dicularly to the observation axis. L1 and L2 are lenses. A is an aperture. E is a very small mirror, and B/S is a 50/50 beam splitter.

250 C. P�eerez L�oopez et al. / Optics Communications 203 (2002) 249–253

spectively. As usual, when the reference and objectwaves interfere, on the CCD sensor, the resultingintensity for the digital hologram corresponding toobject position 1 is

I1ðxc; ycÞ ¼ jrðxc; ycÞj2 þ jo1ðxc; ycÞj2

þ rðxc; ycÞo�1ðxc; ycÞ þ r�ðxc; ycÞo1ðxc; ycÞ;ð1Þ

where the symbol * denotes the complex conjugateof the variable. The last term in Eq. (1) containsinformation about the amplitude and phase of theoriginal object wave. In fact, when Eq. (1) ismultiplied by the reconstruction wave rðxc; ycÞ, asin the classical reconstruction process used inconventional holography, the last term of theequation becomes a replica of the object modu-lated by a term that depends on the reference waveintensity. An equation similar to the former onedescribes the intensity of the digital hologram re-corded for object position 2

I2ðxc; ycÞ ¼ jrðxc; ycÞj2 þ jo2ðxc; ycÞj2

þ rðxc; ycÞo�2ðxc; ycÞ þ r�ðxc; ycÞo2ðxc; ycÞ:ð2Þ

The difference between Eqs. (1) and (2) is that inthe latter a phase term, DW corresponding to theobject rotation, is added in the object wave o2.This phase term is formed by a phase term fordeformation only, Du, and a phase term due to theobject rotation, q, i.e., DW ¼ Du þ q. At thisstage, and for the method proposed here, a Fouriertransform is applied to Eqs. (1) and (2) in order torecover the object amplitude and phase informa-tion [11]. After the phase distribution of each ho-logram is evaluated, both are digitally subtractedresulting in a phase map that directly gives detailedinformation of DW, i.e., object deformation Duand rotation q, between the positions 1 and 2.The object Fourier transform has the form

Oðxf ; yfÞ ¼Z 1

�1

Z 1

�1oðxc; ycÞ

exp½�i2pðxcxf þ ycyfÞ�dxc dyc; ð3Þ

where ðxf ; yfÞ is a point in the Fourier plane, and theterm expð�iDuÞ corresponding to object deforma-tion has been incorporated into the object function

oðxc; ycÞ. Assume that the object rotates anti-clock-wise by a very small angle a. A point on the object,as seen by the CCD sensor, may now be representedin the rotated co-ordinate system (x0c; y

0c) as

xc ¼ x0c cos a � y0c sin a;

yc ¼ x0c sin a þ y 0c cos a:ð4Þ

The rotated object Fourier transform takes theform

ORðxf ; yfÞ ¼Z 1

�1

Z 1

�1oðx0c; y0cÞ

exp½�i2pððx0c cos a � y 0c sin aÞxfþ ðx0c sin a þ y0c cos aÞyfÞ�dx0c dy0c: ð5Þ

Rearranging the argument in the exponential al-lows us to write this equation as

ORðxf ;yfÞ¼Oðxf cosaþ yf sina;�xf sinaþ yf cosaÞ;ð6Þ

which shows clearly that a rotation in the objectspace corresponds to a rotation in the Fourierspace, viz., compare Eqs. (3) and (6).As usual, by expressing Oðxf ; yfÞ and ORðxf ; yfÞ

in their amplitude and phase terms Eq. (6) be-comes

ORðxf ;yfÞ ¼ jOðxf cosaþ yf sina;�xf sinaþ yf cosaÞj exp½i2pðxf cosaþ yf sina;

� xf sinaþ yf cosaÞ�: ð7Þ

Using the assumption that a is small us allows towrite Eq. (7) approximately as

ORðxf ; yfÞ ¼ jOðxf ; yfÞj exp½i2pðxf cos a þ yf sin a;

� xf sin a þ yf cos aÞ�; ð8Þ

where the fact that the Fourier transform of theobject amplitudes is similar, before and after ro-tation, has been used. The latter was demonstratedexperimentally by subtracting the magnitudes ofboth amplitudes. The angle of rotation a is cal-culated using the experimental conditions as in theset-up, and this value is used to obtain the result-ing derotated Fourier spectrum, given by Eq. (8),which is then pass-band filtered and then inverseFourier transformed. The latter two operations areapplied to the unrotated hologram. Finally bothphases are compared by digital subtraction. The

C. P�eerez L�oopez et al. / Optics Communications 203 (2002) 249–253 251

resulting phase map contains data that separatethe deformation Du.In summary, a co-ordinate change due to the

object rotation in only one of the digital holo-grams may be seen as a phase change that does notaffect the object amplitude distribution oðxc; ycÞ,thus making it possible to subtract from DW thephase term q, the object rotation. When the latteris done and the individual digital hologram phasesare subtracted, the phase term Du, the object de-formation, remains, making it possible to quanti-tatively evaluate the object deformation. Thephase term q may be determined from the objectangular speed and the time separation between thedigital hologram acquisition. It should be pointedout here that the object angular displacement a issmall, in such a way as to consider the rotationradial component as a constant.

3. Experimental set-up and results

An interferometric arrangement, as that com-monly used in digital holography for out-of-planesensitivity, is shown in Fig. 1. The object understudy is an 8 cm diameter 4 rigid-blade air fan,rotating at 2600 rpm. The rotating speed chosen iscomparable to the one employed in turbines usedfor power generation.The blades have plane, not curved, surfaces that

are all considered perpendicular to the CCDcamera line of observation, going from the CCDsensor center to the object geometrical center. Theblade rigidity is expected to bring a more signifi-cant component of the rotating motion than thatattributed to the deformation only. Pulses from aruby laser, k ¼ 0:694 lm, are used to illuminatethe rotating fan, which is being imaged through acamera lens attached to a high resolution CCDcamera, 1200 by 1280 pixels and 12 Bit resolution.The object-illuminating beam is co-linear to thefan rotation axis, whose origin coincides with theCCD sensor center. The fan rotating axis is set atan angle h ¼ 10 mrad with respect to the CCD lineof observation, therefore, allowing for a fine ad-justment in the object illumination direction insuch a way that up to three fringes are seen on theDW phase map. This object illumination angle h is

achieved by using a very small mirror, E in Fig. 1,which is on the CCD line of observation. Theaverage size of an individual speckle was set toapproximately four pixels, i.e., 28 lm. Two imageplane digital holograms were captured, with apulse separation of 20 ls, using a typical speckledata acquisition program. As a result of imple-menting the described Fourier transform process,that incorporates the artificial subtraction of the qphase term at the appropriate stage, a phase mapis obtained. Fig. 2(a) shows the wrapped phase

(a)

(a)

Fig. 2. Wrapped phase maps on the four fan blades working at

2600 rpm. (a) The fan blades show the term Dw and (b) the fanblades have the information due only to the deformation con-

tribution from the rotating motion.

252 C. P�eerez L�oopez et al. / Optics Communications 203 (2002) 249–253

DW, with fringes due to the rotational motion andthe deformation due to this rotation. Notice thatno phase fringes are seen out of the fan blade ge-ometry, since there was not another object to beseen. However, the CCD camera sees an obstruc-tion at the center of the fan due to mirror E, adetailed seen in Fig. 2(a). Next, the procedure setforth after Eq. (8) is applied with a ¼ 0:0054 rad,which comes from the above data. The result maybe seen in Fig. 2(b), where the subtraction of theangular displacement phase compensating com-ponent can be clearly appreciated as diagonalfringes surrounding the fan blades, which nowexhibits a phase map related exclusively to its de-formation Du.Within the 20 ls time lapse used for this par-

ticular experiment the result shown in Fig. 2(b)depicts an out-of-plane deformation on the fanblades. The phase map is grey level encoded, andshows a maximum deformation of :35 lm, whichcorresponds to k=2 in Fig. 2(b). This result may becompared to the one obtained in [8], where anoptomechanical derotator was used in an experi-ment where the fan blades were rotating and un-dergoing a deceleration. In that case the maximumdeformation was about 5 lm. The comparisonbetween the two experiments is valid since in bothcases the laser pulse separation was 20 ls, whichmeans that speckle decorrelation is avoided, a mustwhen speckle interferometric techniques are used.

4. Conclusions

The Fourier transform phase decoding tech-nique has been used as a means to create a newoptical method that completely removes the in-herent object rotational movement present duringdynamical deformation studies, thus avoiding theuse of the optomechanical derotator, that is diffi-cult to set up and is very expensive. Furthermore,it is well known that the optomechanical derotatormakes use of a mechanical temporal synchronisa-tion, something not needed in our method wheresynchronisation is made only by the firing of thetwo laser pulses at any object rotation position.The new method is tested in digital holography,where the compensation for the object rotation is

done through a digital phase term subtraction,without affecting the amplitude and phase of theobject deformation. The problem of speckle dec-orrelation between the capture of the two digitalholograms is solved by adjusting the speckle sizeon the CCD sensor, and also by setting the rota-tion speed in such a way that up to three fringesare observed due to the object rotation. Finally,the extension to 3D deformation measurementsmay be attained by using other techniques, such asthe one reported in [12], where three different il-lumination directions are used in order to fullydescribe, in conjunction with object shape, theobject x; y; z deformation components. This will bedone in the near future.

Acknowledgements

The authors wish to thank the financial supportof Consejo Nacional de Ciencia y Tecnolog�ııa(CONACYT), M�eexico, for supporting this re-search through grant 32709-A.

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