Optics Communications 282 (2009) 4595–4601

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Optics Communications

journal homepage: www.elsevier .com/locate /optcom

Off-axis low coherence interferometry contouring

Yves Delacrétaz a,*, Nicolas Pavillon a, Florian Lang b, Christian Depeursinge a

a Ecole Polytechnique Fédérale de Lausanne (EPFL), Advanced Photonics Laboratory, CH-1015 Lausanne, Switzerlandb Centre Hospitalier Universitaire Vaudois (CHUV), Service ORL, Rue du Bugnon 21, CH-1005 Lausanne, Switzerland

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 May 2009Received in revised form 18 August 2009Accepted 21 August 2009

PACS:42.25.Hz42.25.Kb07.60.Ly

Keywords:InterferometryLow coherenceContouringOff-axis

0030-4018/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.optcom.2009.08.048

* Corresponding author. Tel.: +41 216935182; fax:E-mail address: yves.delacretaz@epfl.ch (Y. DelacrURL: http://apl.epfl.ch/muvision.html (C. Depeursi

In this article we present a method to achieve tri-dimensional contouring of macroscopic objects. A mod-ified reference wave speckle interferometer is used in conjunction with a source of reduced coherence.The depth signal is given by the envelope of the interference signal, directly determined by the coherencelength of the source. Fringes are detected in the interferogram obtained by a single shot and are detectedby means of adequate filtering. With the approach based on off-axis configuration, a contour line can beextracted from a single acquisition, thus allowing to use the system in harsh environment.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

There is a large panel of techniques available for optical 3Dmeasurements of rough or diffusing macroscopic objects with asub-millimeter accuracy [1]. Incoherent light can be used withfringes projection techniques, where the fringes deformation isconsidered in order to retrieve the depth information. Moiré pat-tern techniques also exist, in which two gratings are used to gen-erate contour fringes [2]. Photogrammetry employing mainlystereo vision procedures to obtain tri-dimensional shape are alsoavailable [3].

Interferometric measurements are widely used to obtaincontouring of macroscopic objects, in particular the so-calledelectronic speckle pattern interferometry technique (ESPI). Jaisinghand Chiang obtained the surface of a regular light bulb [4], using asetup with double exposure. Joenathan et al. used a setup contain-ing a Fourier-filtering part to extract the information from a spec-klegram [5]. Rodriguez-Vera et al. have used a fibered setup forout-of-plane sensitive ESPI [6]. Prieto and Garcia-Sucerquiareported contouring of macroscopic objects with a phase-differ-ence method [7]. All these techniques require at least a doubleexposure procedure. Balboa et al. reported 3D measurements using

ll rights reserved.

+41 216933701.étaz).nge).

superluminescent diode and multimodes laser diodes in an opticalfiber based interferometer [8], using a five step algorithm for fringeamplitude extraction. One way to access tri-dimensional measure-ment is to achieve phase change by using at least two wavelengths,as Tatam et al. reported [9], or a source emitting an extended spec-trum. This gives rise to the Fourier-transform speckle profilometry[10]. Another possible method for phase extraction from a speckle-gram is to use spatial phase shifting (SPS), in which the phase of apixel is estimated from the intensity of the surrounding pixels, in-stead of the values of the same pixel at different times, as it is thecase for time phase shifting. Bhaduri et al. achieved SPS with theuse of a double aperture mask [11]. In order to overcome the lim-itation of multiple acquisitions, many efforts were done to buildinterferometers allowing to record multiple interferograms at thesame time. Recently, Hrebesh et al. proposed a system for theacquisition of three phase-stepped interferograms and a referenceimage at the same time, allowing single-shot low-coherence time-domain profilometry [12].

When using a broadband light source in a typical out-of-planesensitive ESPI interferometer, the process of extracting the locationof the coherent superposition of two waves is referenced in the lit-erature as the ‘‘coherence radar” technique. Dresel et al. used asimple Michelson configuration, with a piezo-electric actuator totake three phase-stepped acquisitions in order to extract a contourdepth [13]. It is also possible to use short light pulses, in this casethe method is called ‘‘light-in-flight holography” [14]. Carlsson

4596 Y. Delacrétaz et al. / Optics Communications 282 (2009) 4595–4601

et al. further modified the technique using a blazed grating tointroduce a spatially varying delay across the reference beam pro-file [15,16].

In this paper a novel arrangement for low coherence interfer-ometry is presented. Similarly to the coherence radar technique,the goal is to achieve contouring by isolating the zone in the ac-quired image where fringes are created. It will be shown that, witha well designed off-axis configuration, it is possible to achieve con-tour extraction in only one acquisition, removing the need forphase-shifting or multiple acquisitions with different wavelengths.First, the setup and the procedure used for fringes extraction arecarefully described. The optical system has been totally simulated,and results of the simulation are compared with measurements.They show fair agreement. Finally, different results are presented.Namely, the first on a tilted plane used for technique validationand precision evaluation, the second one on the inner side of a cyl-inder, and the last on a wooden pencil tip.

2. Material and method

2.1. Setup

We achieve curve level extraction with the use of the modifiedsmooth reference wave speckle interferometer depicted in Fig. 1.The source is a reduced coherence length laser diode (LD), with acoherence length of about 0.3 mm. A beam splitter (BS1) separatesthe collimated beam generated by the LD into the reference beam(R) and the object beam (O). The reference beam is directed to a de-lay stage (DS), composed of two mirrors (M1 and M2), mounted ona motorized axis. A lens (L1) is placed in the object arm of the inter-ferometer. It is used to widen the illuminated area of the object andto form an image at the CCD plane. The adjustable diaphragm (D) isan essential part of the setup: it is used to limit the aperture of thesystem, thus making possible to create interferences composed offringes modulated by a slow varying speckle pattern. L2 is usedto adapt the spherical reference wave in order to match the illumi-nation geometry.

The intensity of two waves with finite temporal coherence canbe expressed as [17]:

I ¼ 2I0 1þ g sð Þj j cos / sð Þ½ �; ð1Þ

where gðsÞ is the complex degree of temporal coherence, with s thedelay time between the two waves. / is the phase difference in-duced by s. Eq. (1) expresses the ability of a wave to interfere with

Fig. 1. Experimental interferometric setup used. LD: laser-diode with low coher-ence length. BS1, BS2: beam splitters. L1, L2: Lenses. D: adjustable diaphragm. Obj:scattering sample. DS: delay stage. CCD: charge-coupled device camera.

a time delayed version of itself. For a light source with a bandwidthof Dm, the visibility of the fringes for two waves delayed bysc � 1=Dm is 0.5. The path that light has traveled during sc is calledthe coherence length (Lc). Therefore, interference terms will bepresent only on points in the image for which the optical path dif-ference (OPD) between R and O is below Lc . Extracting from the im-age the positions where these interference fringes are situated givesa depth signal for macroscopic objects (with height much greaterthan Lc), and an easy contouring procedure by either shifting R orO. On the setup depicted on Fig. 1, the DS is used first to adjustthe length of the reference arm in order to match the length ofthe object arm, then the object is scanned to retrieve different curvelevels.

M2 is placed so that there is a small angular mismatch at BS2between the optical axis of R and O, which in turn can be regardedas interferences of spherical waves coming from off-axis points.This creates ‘‘carrier fringes” on the interferogram. In the Fourierspectrum, this separates the interference terms and the zero-orderterm. Detecting the presence of these fringes with a local Fourierspectrum evaluation becomes then possible and provides a simpleand efficient way of extracting the area on the image where coher-ent superposition of the two waves occurs, thus making possible ofextracting a curve level in only one acquisition.

2.2. Contour lines extraction

As pointed out in the previous subsection, it is possible to createfringes that are discernible with a small angular mismatch be-tween the optical axis of R and O. Extracting the area where thereare fringes could also be achieved by phase shifting manipulation.However, this would involve the acquisition of at least two inter-ferograms, thus increasing the overall acquisition time and thesensitivity to perturbations.

Common procedure to extract fringes in an interferogram is tosimply process its Fourier transform with adequate bandpass filter.However in our case this approach fails, mainly due to the fact thatthe fringes to extract may be very low contrasted and are highlylocalized, so that after numerical calculation of the Fourier trans-form, the magnitude of the spatial frequencies in the bandpass fil-ter are approaching the noise limit.

The interferogram is here processed with a window filter, de-fined by evaluating the Fast Fourier Transform (FFT) in aneighborhood.

The output value of the pixel that is processed is the maximalamplitude of the local spectrum, after zero-order removal:

yðp; qÞ ¼ maxm;n2½1;M�1�

hm;n

�� ��; where :

hm;n ¼XM�1

k¼0

XM�1

l¼0

xðpþ k; qþ lÞ � e�iðmw0ðpþkÞþnw0ðqþlÞÞ;

with : p; q 2 ½0;N � 1�: ð2Þ

Eq. (2) corresponds to a local measurement of the high frequenciescontribution, with xðp; qÞ the input interferogram, w0 the slowestpulsation considered, hm;n the window filter and yðp; qÞ the filteroutput value. It is also possible to evaluate the mean value or thecumulative sum of a certain bandwidth of spatial frequencies. Itwas observed that the output of the maximum value is sufficientlyselective, providing that there is no sinusoidal variation of theintensity induced by the object itself that falls into the frequencycalculated by the FFT.

Fig. 2a shows the obtained signal for the extraction of ameridian on a metallic cylinder. Fig. 2b and c shows the differentcomponents of the FFT (after zero-order removal) calculated overa neighborhood of 8� 8 pixels, when the pixel of interest be-longs to an area where OPD is smaller than Lc and greater than

Fig. 2. (a) Extracted signal using Eq. (2) on a metallic cylinder with a source with Lc ¼ 0:3 mm. (b) and (c) are the local Fourier spectrum (after zero-order removal) for a pixelinside and outside the curve to be detected, respectively.

Y. Delacrétaz et al. / Optics Communications 282 (2009) 4595–4601 4597

Lc , respectively. As it can be seen from Fig. 2b and c, the maxi-mum of the local Fourier spectrum evaluation is 3 times strongerwhen a pixel is situated in a zone where coherent superpositionoccurs.

Fig. 3. Simulation of the imaging part of the setup in Fig. 1. do and di: object-lens

3. Interferogram simulations

The imaging part of the optical system of Fig. 1, consisting of theimaging lens L1 and the diaphragm D are approximated by a thinlens and a small aperture, as shown on Fig. 3. Given an optical fieldU0ðx0; y0Þ in the object plane, the optical field at the apertureUpðx; yÞ can be easily calculated with the convolution formulationof the free space propagation in the Fresnel approximation. Its dis-crete formulation for numerical implementation can be expressedas [18]:

and lens-image distances, respectively. R: radius of the diaphragm used to limit theaperture of the optical system. U0: input object optical field, Up: field afterpropagation over a distance d0 (before the diaphragm), Uf : field after thediaphragm/lens system, U1: image field on the CCD plane (propagation over a

Upðm; nÞ ¼1

ikdoe

i2pdok � FFT�1 FFT U0f g � e�ipkdo p2

kþp2

lð Þn o

m;n; ð3Þ

distance di).

where pk ¼ k=ðNDxÞ; pl ¼ l=ðNDyÞ. N is the number of samplingpoints, Dx and Dy are the sampling steps for directions x and y,respectively. It is important to notice that in the convolution for-malism, the sampling steps are the same for the input plane andthe output plane.

The influence of the small aperture can be efficiently calculatedby multiplying the field after propagation with a circular functionin the reciprocal Fourier domain. The lens bends the field U0pðx; yÞwith a quadratic phase term, and the field Uf ðx; yÞ right after the

lens is calculated with the thin lens formula. The field U1ðx; yÞ isthen obtained by propagating the field Uf ðx; yÞ using Eq. (3) replac-ing do by di. The distances do and di are chosen so that the CCDplane corresponds to the image plane, thus satisfying the condition1=do þ 1=di ¼ 1=f .

Considering coherent light backscattered by a rough surface, theelectrical field at a point in space can be seen as the sum of all thecontributions of the illuminated points. Assumption is made that

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the amplitude and phase of each contribution are independent be-tween them. When the roughness is greater than the wavelength,the different phases contributions are uniformly distributed be-tween 0 and 2p.

Based on these considerations, the speckle pattern induced bythe illumination of the object was simulated by adding a randomphase term uniformly distributed on ½0;2p� to the field U0ðx; yÞ,on the object plane. Starting from the hypotheses above, Goodmanderived the probability density function for the real and imaginaryparts of the field [19], and hence, the probability density functionfor the intensity and the phase is:

pðIÞ ¼1Ih i e� I

Ih i; I P 0;

0 ; otherwise;

(ð4aÞ

pðhÞ ¼1

2p ; h 2 �p;p½ �;0; otherwise:

(ð4bÞ

The statistic of the intensity of an image acquired without referencewave when observing a flat sheet of paper is compared with Eq.(4b), and the result is presented in Fig. 4. Crossmarks are the densityvalues obtained from the histogram of the acquired image. Theplain curve is the probability density function calculated with Eq.(4b). Measurements on the acquired image are in fair agreementwith the analytical formulation, which proves that the introductionof a random phase term in the input field is a realistic process tosimulate the speckle induced by the backscattered field. Small dif-ferences between the computed PDF and the measured one cannevertheless be observed. A shift toward smaller intensities isclearly visible on the measured PDF, which could be attributed tothe fact that the illumination is not completely uniform for oursample, and also to the dark regions on the image due to thegrid-lines printed on the papersheet used as a sample. It has to benoted that the speckle pattern depends on the type of object underinvestigation. For example, materials with a roughness of the orderof the wavelength or greater suit well to the uniformly distributedphase assumption. On the other hand, smoother objects would cor-respond to a PDF with its maximum probability shifted towardhigher intensities, hence in this case the intensity distribution couldnot be described anymore with Eq. (4b).

Finally, on the CCD plane, the object wave U1ðx; yÞ interfereswith a smooth spherical reference wave.

All the formalism above is developed using only a single valuefor wavelength, corresponding to pure monochromatic light. TheLD used in the setup described in Fig. 1 has an emission spectrum

Fig. 4. Probability distribution function (PDF) of the speckle intensity in the imageplane. Crosses: evaluation on the acquired image, curve: PDF calculated using Eq.(4b). Inset: acquired speckle pattern at the CCD plane.

whose width is about 1.5 nm. The superposition of R and O is thuspartially coherent. The intensity resulting from the interference ofthe two partially coherent waves is calculated by:

Iðx; yÞ ¼ U1ðx; yÞj j2 þ Urðx; yÞj j2

þ 2 U1ðx; yÞj j Urðx; yÞj j g1rðx; yÞj j cosðDuÞ; ð5Þ

where Du ¼ argfU1ðx; yÞg � argfUrðx; yÞg and g1r is the complex de-gree of temporal coherence between the object and the referencewaves, also called the normalized autocorrelation function. To inte-grate into the simulation the effect of the reduced coherence,knowledge of the width of the emission spectrum is sufficient.jg1r j is approximated by a Gaussian function with full width at halfmaximum (FWHM) equal to Lc . The coherence length of the sourcecan be deduced from the width of the emission spectrum with theevaluation: Lc � k2

0=Dk, and its value corresponds to the physicallimit of the contouring axial resolution that can be achieved withthe device. Thus, in order to obtain valuable results, height varia-tions of the object must be larger than Lc .

By introducing the partially coherent nature of the light emittedby the laser diode at the end of the simulation, the speckle patternis considered as if created by a totally coherent illumination. Errorsintroduced by this approximation are nevertheless negligible. In-deed, taking into account a single plane as input for the simulation,the optical path difference induced by each scatterer considered asa source of a small spherical wave contributing to the creation ofthe speckle pattern on the object is considerably smaller than Lc.

Fig. 5 shows the interferograms obtained from the simulation(left side), along with experimental verifications obtained withthe setup depicted on Fig. 1 (right side), each time the Fourierspectrum is presented as an inset. Parameters for the simulationare: k ¼ 661 nm, do ¼ di ¼ 10 cm, Dx ¼ Dy ¼ 4 lm, N ¼ 1024,x0 ¼ 600 lm, y0 ¼ 200 lm, z0 ¼ 47:35 mm, with ðx0; yo; z0Þ the ori-gin of the spherical reference wave. Fig. 5bb and d were obtainedby observing a flat lightly diffusing object (a sheet of standardwhite paper) perpendicular to the optical axis. The amplitude ofthe input field for the simulation is constant (jU0j ¼ 1), and a uni-formly distributed random phase term was added to simulate forthe roughness of the object. To obtain results on Fig. 5aa and c, Rwas set to 0.3 mm and 1.5 mm, respectively. Experimental resultson Fig. 5b and d were obtained for aperture diameters of about1 mm and 5 mm, respectively.

As in off-axis holography [20], the introduction of carrier fringesseparate the two image terms (R�O and RO�) and the zero-orderterm (jOj2 þ jRj2) in the Fourier spectrum. Due to the roughnessof the sample, there is an almost homogeneous contribution ofall the spatial frequencies, up to the cutoff frequency of the opticalsystem, which is in turn determined by the aperture size. A systemwith a large aperture will create a fine speckle pattern (Fig. 5c andd). For plane waves, if the angle between the reference and the ob-ject wave is not large enough, the interference terms will thenoverlap with the zero order term in the Fourier spectrum. Forspherical waves, it is the relative distance between the origin ofthe waves that is to be considered. If the speckle pattern is finerthan the fringes periodicity, no modulation will be distinguishable.On the other side, for each points on the detector, the angle be-tween the propagation vector of the reference wave and the objectwave must be below 3 degrees, this limit is imposed by the pixelsize of the CCD to guarantee adequate sampling of the fringes [21].

On real signals, there is most often not a high contrast enhance-ment between the zone where coherent superposition of R and Ooccurs and the incoherent area of the signal. Fig. 6a and b presentsimulated interferograms obtained with the same parameters asthose used to obtain Fig. 5a and c, except that the coherence lengthwas shortened down to 10 lm and that the ratio between the ref-erence and object waves was adapted in order to create fringes

Fig. 5. (a) Simulated interferogram with a diaphragm diameter of 0.6 mm. (b) Experimental interferogram with a diaphragm diameter of 1 mm. (c) Simulated interferogramwith a diaphragm diameter of 3 mm. (d) Experimental interferogram with a diaphragm diameter of 5 mm. Corresponding Fourier spectrum are presented as insets.

Y. Delacrétaz et al. / Optics Communications 282 (2009) 4595–4601 4599

with a low contrast. Fig. 6a and b were obtained with a simulatedoptical system having an aperture diameter of 0.6 mm and 3 mm,respectively. On Fig. 6c and d are shown the results of the filteringprinciple following Eq. (2). Points on the image for which OPD isbelow Lc are correctly isolated on Fig. 6, but it is not the case onFig. 6.

The speckle size is a key factor, for if the speckle size is of thesame size as the fringes periodicity that we want to detect, the ex-tracted meridian will be highly noisy. The speckle size is inverselyproportional to the aperture size. This imposes an upper limit onthe aperture of the optical system, and thus a limit on the lateralresolution. This means in turn that the mismatch between the opti-cal axis of R and O, and the diameter of the diaphragm have to bechosen so that the fringes can be resolved, and at the same timethat the mean speckle size is sufficiently large in order to achievegood meridian extraction.

This shows that in order to achieve contouring with our tech-nique, the diaphragm of the imaging system included in the inter-ferometer has to be reduced down to the point where speckle sizeis larger than the fringes created by the off-axis geometry, so thatthe interferometric carrier fringes can spatially modulate thespeckle pattern. It has been observed that a mean speckle size ofabout 2 times the periodicity of the fringes is adequate, whichcomplies with the Nyquist–Shanon condition. Consequently, thelower limit for the speckle size is 4 pixels, since it is not possibleto create carrier fringes with a periodicity smaller than 2 pixels.Moreover, the local window used for fringes extraction must belarger than the speckle size. Good results are obtained with a win-

dow filter size of 8� 8 pixels, allowing a trade-off between compu-tational complexity and sampling of the different spatialfrequencies.

4. Experimental results

The first measurement has been achieved on a tilted plane.Scanning has been performed by translating M1 and M2 mirrorsof Fig. 1, with steps of 1 mm. A threshold has been applied onthe filtered results. 3D positioning of all the extracted contour linesis represented in Fig. 7b. Fig. 7a shows a typical interferogram, witha zoom on fringes as inset, along with the results of the filtering asdescribed in Eq. (2), before thresholding. Measurement has beenrepeated for different orientations. Root mean square error (RMSE)between the points actually on the plane and the measured coor-dinates is 0.4 mm. Let us recall the coherence length of the sourceof 0.3 mm, which represents the physical limit for the accuracy ofthe method. Comparing this two values, it appears that the errorintroduced by the fringes extraction procedure is acceptable formacroscopic objects contouring.

Example of measurement on the inner side of a 13 mmdiameter cylinder made of standard white paper is presented inFig. 8. The extracted curves are superimposed to a classical image.To obtain these contours, the DS was translated by 8 mm, which isalso the height difference between these two isovalue lines. Due tothe inherent magnification dependence on the z position of the ob-ject, the meridians seem to have different diameters. By measuring

Fig. 6. (a) and (c): Simulated interferogram and extracted signal with short coherence length and diaphragm diameter of 0.6 mm. (b) and (d): Simulated interferogram andextracted signal with short coherence length and diaphragm diameter of 3 mm.

a

0200

400600

800

0200

400600

8000

5

10

15

20

x (px)y (px)

z (m

m)

b

Fig. 7. Tilted plane measurement. (a) Typical signal, zoomed fringes as inset, alongwith the output of the ‘‘LocalFFT” filter before thresholding. (b) 3 D view of severalextracted level curves.

Fig. 8. Two level curves extracted on the inner side of a 13 mm diameter cylinder.Contour lines are separated along the z-direction by 8 mm.

4600 Y. Delacrétaz et al. / Optics Communications 282 (2009) 4595–4601

the transverse magnification as a function of the object position,true tri-dimensional positioning of the contour lines can beachieved. An alternative could be to use object space telecentricoptics, as they are not affected by object position magnificationchange. A drawback of telecentric optics is that they have to beas large as the object under observation.

Fig. 9 shows the results obtained when observing the woodentip of a colour pencil. Interferograms were acquired by translatingthe DS by steps of 0.1 mm. After contour line extraction, a set ofabout 75’000 3D coordinates are represented. In order to obtainreal coordinates related to the object referential, the z-dependency

Fig. 9. Acquisition of about 75’000 3D points on a wooden pencil tip. Inset:snapshot of the pencil tip, with the black box delineating the measured area.

Y. Delacrétaz et al. / Optics Communications 282 (2009) 4595–4601 4601

of the magnification has been measured. With the help of this cor-rection factor, tri-dimensional positioning of the extracted pointsexpressed in physical units is possible.

5. Conclusion

In this article we proposed a new method for optical contouringof macroscopic objects. With this system, based on a modifiedsmooth reference wave speckle interferometer using a reducedcoherence laser diode, it is possible to extract a contour line with

only one acquisition. The system has been completely simulated,with an emphasis on the importance of the size of the diaphragm.Results were presented on different samples, especially an evalua-tion of the accuracy of the method has been achieved by tilting aplane and calculating the RMSE of the measurement. It showedup a precision of 0.4 mm.

Acknowledgement

The authors are thankful of the Swiss National Science Founda-tion for supporting part of this work through research Grant 3200-109916.

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