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2872 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 Barbosa et al.

Single-exposure, photorefractive holographicsurface contouring with

multiwavelength diode lasers

Eduardo A. Barbosa and Antonio A. V. Filho

Laboratório de Óptica Aplicada, Faculdade de Tecnologia de São Paulo, Centro Paula Souza,Universidade Estadual Paulista, Pça Cel Fernando Prestes, 30, CEP 01124 060, São Paulo–SP, Brazil

Marcos R. R. Gesualdi

Instituto de Física, Universidade de São Paulo, CP 66318, CEP 05315-970, São Paulo–SP, Brazil

Bruno G. Curcio

Laboratório de Óptica Aplicada, Faculdade de Tecnologia de São Paulo, Centro Paula Souza,Universidade Estadual Paulista, Pça Cel Fernando Prestes, 30, CEP 01124 060, São Paulo–SP, Brazil

Mikiya Muramatsu and Diogo Soga

Instituto de Física, Universidade de São Paulo, CP 66318, CEP 05315-970, São Paulo–SP, Brazil

Received February 23, 2005; revised manuscript received April 25, 2005; accepted May 31, 2005

We studied the use of multiwavelength diode lasers for surface profilometry through holographic recording insillenite Bi12TiO20 crystals. When such lasers are used, the holographic image from single-exposure recordingsappears covered with interference fringes providing information on the surface relief of the object. By takingadvantage of the narrow interference fringes due to the multiwavelength emission of the laser, we obtainedinterferograms by holographic recording with two reference beams, which improves the surface analysis byvisual inspection and enhances the profilometry sensitivity. © 2005 Optical Society of America

OCIS codes: 120.2880, 120.2650, 120.6650.

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. INTRODUCTION

urface contouring, or profilometry, has a wide range ofpplications in areas such as manufacturing quality con-rol and machine vision.1–3 The optical contouring meth-ds have the additional advantage of providing nonde-tructive processes, and among them the holographic onesllow for very precise, whole-field measurements withasy qualitative visual inspection.4,5

Holographic contouring is generally based on two-xposure, two-wavelength recording; i.e., in each expo-ure the holographic image is recorded with a differentavelength. In the holographic reconstruction the result-

ng interference pattern is the contour map of the object.ecently the formation of interference contour fringes

hrough single-exposure recording in sillenite crystals bysing multiwavelength, large-free-spectral-range (FSR)iode lasers was demonstrated.6 It was shown that theiffracted light intensity strongly depends on the phaseifference between the reference and the object beamsnd on the laser FSR. In single-exposure processes, theolographic image appears covered with interference

1084-7529/05/122872-8/$15.00 © 2

ringes corresponding to the contour lines of the objecturface. By phase-shifting the reference beam, the fringesun along the surface, thus allowing for quantitativehree-dimensional (3D) scanning of the object.7

The use of lasers emitting more than two longitudinalodes results in bright interference fringes which arearrower than those of conventional two-wavelengthechniques.6 In this work we use this characteristic to ob-ain holographic images with two properly shifted refer-nce beams. In this case, the resulting interferogramsave twice the spatial frequency compared with the one-eference-beam case. This procedure allows for a more ac-urate visual inspection and provides less noisy measure-ents.In the interferogram quantitative analysis, the phaseapping was performed through the phase-stepping tech-ique (PST)8 and the phase unwrapping was carried outhrough the cellular-automata method (CAM).9,10 In therst, we employed PST in a four-frame process by apply-

ng a � /2 phase shift between each interferogram. In or-er to obtain simple, analytic expressions for the phaseap we considered the laser emitting in three modes of

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Barbosa et al. Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2873

qual intensity. The validity of this assumption is dis-ussed by comparing the holographic contouring resultsith those obtained by conventional measurements.

. HOLOGRAPHIC RECORDING WITHULTIMODE LASERS

. Holographic Imaging with One Reference Beamet us consider the incidence of the reference (R) and thebject (S) waves of wavelength � onto a photorefractivei12TiO20 (BTO) crystal. Since both beams originate frommultimode laser, the beam amplitudes RN and SN at theTO input can be written as

RN�0� = R0 �n=−�N−1�/2

n=�N−1�/2

An exp�i��k + n�k��R + �n��,

SN�0� = S0 �n=−�N−1�/2

n=�N−1�/2

An exp�i��k + n�k��S + �n��, �1�

here N is the number of oscillating modes, k�2� /� ishe wavenumber, �k is the wavenumber interval betweenwo adjacent modes, �S and �R are the optical paths of thebject and the reference beams, respectively, and �n is thehase of the nth mode at the laser output. The coefficientn is real and R0

2+S02 is the incident light intensity. Con-

ider the hologram recorded purely by diffusion with therystal cut in a (110) transverse electro-optic configura-ion. If the readout process is accomplished by self-iffraction, the intensity ID of the reconstructed objectave is given as a function of the diffraction efficiency �y

ID = �IR � m2IR = 2 RN* SN

RN2 + SN2�IR, �2�

here m is the interference pattern visibility and IR is theeference beam intensity. The superscript * refers to theomplex conjugation of the wave amplitude. Since there iso mutual coherence between different modes, the phaseifference �n−�m between them depends randomly onime. For this reason, the interference of different modesas no effect on the holographic imaging. The intensity ofhe holographic reconstruction of the object beam canherefore be written with the help of Eqs. (1) and (2) as6

ID � 2m0� sin�N�k��S − �R�/2�

sin��k��S − �R�/2� 2

IR, �3�

here m0�2R0S0 / �R02+S0

2� and An=1 in Eq. (1) for sim-lification. In relation (3) the terms containing �n−�mere neglected. By examining this relation the formationf interference fringes on the holographic image becomeslear: Since the phase �k��S−�R� depends exclusively onhe surface relief, the interference fringe is the regionhere the distance between the surface and the front facef the crystal is constant. Considering two points A and Bach lying on adjacent bright fringes, one gets from rela-ion (3) the optical path difference between A and B as6

m

�S,B − �S,A =2�

�k=

�2

��� �S, �4�

here �S is the synthetic wavelength and �� is the wave-ength interval between adjacent modes. The term �2 /��s related to the laser FSR �� for longitudinal modes dueo spectral hole burning11 by the expression �2 /��=c /��2L, where L is the laser resonator length. Thus, thehorter the cavity of the laser, the more interferenceringes within a given depth interval on the surface.

For the reasons disclosed above, the typically shortavities of diode lasers make them a very suitable lightource for real-time, single-exposure holographic profilo-etry purposes. In our setup, the beam incident on the

tudied surface and the beam scattered from it propagaten nearly opposite directions. Thus, the depth differencez between two adjacent bright (or dark) fringes will bez���S,B−�S,A� /2=L. For this reason, lasers with, e.g.,0 cm-long resonators would provide a depth difference of20 cm between adjacent fringes. This large depth differ-

nce makes long-resonator lasers not viable for profilom-try by this technique.

. Surface Profilometry by Phase-Stepping Techniqueith One Reference Beamhis technique of fringe evaluation is based on the inten-ity change of the interferogram by phase-shifting one ofhe interfering beams in the holographic setup. In theurrent case, this is carried out in a four-frame process,hrough which the reference beam is sequentially phase-hifted by 0, � /2, �, and 3� /2. Thus, from relation (3) thentensity at a point �x ,y� can be written as

ID,l�x,y� � 2m0� sin�N��k�S�x,y� + l�/2�/2�

sin���k�S�x,y�/2 + l�/2�/2� 2

IR, �5�

here l=0, 1, 2, and 3. Considering N=3 as the number ofscillating modes of the laser used in our experiments,ne gets by simple trigonometric manipulation the rela-ive phase �S�x ,y���k�S�x ,y� /2 of the surface as a func-ion of the interferogram intensities ID0, ID1, ID2, and ID3:

�S�x,y� =1

2arctan ID1 − ID3

ID0 − ID2� . �6�

ig. 1. Interference fringe profile as a function of �S for N=2dotted curve), N=5 (dashed curve), N=8 (solid curve) oscillating

6

odes. In this case, �k=1.39 rad/mm and �=670 nm.

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2874 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 Barbosa et al.

From Eq. (6) one gets a phase map in a gray-level in-ensity, with the range −��S� corresponding to 256ray levels, from black ��S=−�� to white ��S=��.

. Holographic Imaging with Two Reference Beamsrom relation (3) one can see that the bright fringe widthnd the number of oscillating modes are inversely propor-ional. Figure 1 shows the behavior of a bright fringe for=2 (solid curve), N=4 (dotted curve), and N=8 (dashed

urve) modes for �k=1.39 rad/mm around �=670 nm.6

he sensitivity of the surface qualitative and quantitativenalysis can thus be increased by employing a laser emit-ing more than two modes and an optical setup with onebject beam and two reference beams. Let us consider thencidence of two reference beams 1 and 2 with intensitiesR1 and IR2, such that IR1=IR2=IR /2. In this case, the re-ulting object holographic reconstruction is the superpo-ition of two holographic images, each one generated byhe interaction of the object wave with one of the refer-nce waves. Thus, the interferogram on the holographicmage has the following intensity distribution:

ID� � 2m0�� sin�N�k��S − �R1�/2�

sin��k��S − �R1�/2� �2

+ � sin�N�k��S − �R2�/2�

sin��k��S − �R2�/2� �2 IR, �7�

here �R1 and �R2 are the optical paths of the referenceeams 1 and 2, respectively. Each reference beam i con-ributes to the formation of a fringe pattern on the object

ig. 2. Interferogram intensity distribution as a function of surwo reference beams.

ig. 3. Phase distribution as a function of depth z resulting fromeams.

urface whose spatial intensity distribution depends onhe difference �S−�Ri. The holographic image resultingrom the light intensity distribution given by relation (7)as therefore the sum of both interferograms. By adjust-

ng the optical paths of the reference beams so that �R2�R1=q�2 / �2���=q�S /2 �q=1,3,5. . . �, relation (7) re-uces to

ID� � 4m0IR�4 cos2��keff��S − �R1�/2� + 1�. �8�

For convenience we define from the relations above theffective synthetic wavelength as � =�2 / �2���=� /2

pth z for holographic recording with (a) one reference beam, (b)

graphic recording with (a) one reference beam, (b) two reference

ig. 4. Scheme of the optical setup: M1–M5, mirrors; BS1 andS2, beam splitters; L1–L3, lenses; P1 and P2, polarizers; PR1nd PR2, 90° prisms; CCD, camera; PC, computer.

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Barbosa et al. Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2875

nd the effective wavelength interval as �keff�2� /�eff.igure 2(a) shows the intensity distribution of the inter-

erogram generated by a single reference beam for N=3aser modes as a function of the depth z according to re-ation (3), while Fig. 2(b) shows the fringe pattern gener-ted by two reference beams from relation (8) with opticalaths adjusted so that the condition �R2−�R1=q�S eff iset. Both figures have the same z scale. It can be clearly

een from Fig. 2 and from relation (8) that the second in-erferogram has twice the spatial frequency of the first.onsequently, despite the slight decrease of the fringe vis-

bility in the second case, the profilometry sensitivity iswice enhanced when the optical paths of the two refer-nce beams are properly adjusted. Moreover, as demon-trated by Millerd and Brock,4 by lowering the syntheticavelength the noise in the profile measurements is de-

reased.

. Phase-stepping Technique with Two Referenceeamsrom relation (8) the four-frame PST is employed again.n this case, both reference beams are phase shifted by 0,/2, �, and 3� /2 (with respect to �S eff), resulting in four

rames whose intensities are given by

Fig. 5. Surface analysis of a 30°-tilted flat bar: (a) Interfer

ID,l� � 4m0IR�4 cos2���keff�S + l�/2�/2� + 1�, l = 0,1,2,3.

�9�

By performing the same operations as in Subsection.C, one obtains the following expression for the phaseS� �x ,y�:

�S� �x,y� =1

4arctan−

ID1� − ID3�

ID0� − ID2�� . �10�

Figures 3(a) and 3(b) show the dependence of �S�x ,y�nd �S� �x ,y� on the depth z, for �keff=2�k=2.36 rad/mmnd �=670 nm. Both figures have the same z scale. As ex-ected, when two properly shifted reference beams aremployed, the resulting phase map has twice the spatialrequency of the map with only one reference beam.

. OPTICAL SETUPigure 4 shows the setup for the study of the surface relieff a rigid object through holographic recording with aTO crystal. The object beam is collected by the lens L1,hich forms the image at the BTO crystal, while the lens2 builds the holographic object image at the CCD cam-

, (b) phase map, (c) unwrapped phase pattern, (d) 3D plot.

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2876 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 Barbosa et al.

ra. The numerical aperture of the lens L1 is 2.8, so thathe holographic image resolution is limited by the pixelize of the CCD camera �16 m�.

The crystal is cut in a (110) transverse electro-opticonfiguration with the [001] axis orthogonal to the inci-ence plane. The holographic image is displayed in a com-uter monitor for further processing. By using the aniso-ropic diffraction properties of the crystal, the incidenteams are linearly polarized by the polarizer P1 at therystal input, so that the transmitted and the diffractedeams are orthogonally polarized at the BTO output in aiffusion recording regime.12,13 The polarizer P2 can beroperly rotated in order to allow maximum interfero-ram visibility. The beam splitter BS1 splits the incidentaser beam into the object and the reference beam, whileS2 splits the reference beam into the reference beams 1nd 2. The 90°-prism PR1 is attached to a micrometer inrder to adjust the value �R2−�R1. The 90°-prism PR2nd the micrometer supporting PR1 are mounted on aranslation stage which introduces the same phase shiftimultaneously in both reference beams. The use of therisms in both reference beams keeps the superposition ofhe reference and the object beams at the BTO crystalhroughout the phase-stepping process. The paths of theeference beams can be seen in Fig. 5.

We use as light source a 30 mW diode laser emittinground 670 nm with FSR ��=53 GHz, corresponding tok=1.18 rad/mm and �z=2.66 mm. According to the ho-

ographic measurements performed in Ref. 6 this laser

ig. 6. The same analysis as depicted in Fig. 5 for a metallic cyd) 3D plot.

mits in four modes with different intensities, i.e., AiAj in Eq. (1). In order to obtain the analytic solutions for

he phase given by Eqs. (6) and (10), we assume the oscil-ation of three equally intense laser modes, so that the in-ensity of the holographically reconstructed image isiven by relation (3). As will be seen later, this simplifyingssumption leads to very satisfactory results.

. EXPERIMENTS. One Reference Beame first blocked the reference beam 1 and performed a

ingle exposure in order to obtain the holographic imagerom a flat, metallic plate with only the reference beam 2.his object was tilted 30° with respect to the front face of

he BTO. The hologram recording time was �10 s. Themage of the object with the characteristic interferenceontour fringes is shown in Fig. 5(a). In this case, the dis-ribution of the fringes along the object surface is deter-ined by relation (3). From Eqs. (5) and (6) the 3D scan-ing of the object can be accomplished by phase stepping.or each step, the translation stage was displaced byz /4=0.66 mm. The corresponding phase map by thisrocess is shown in Fig. 5(b), while the unwrapped phaseattern obtained through the CAM is shown in Fig. 5(c).he surface relief is depicted in Fig. 5(d), with an en-

arged scale in the z direction. From the data shown in

(a) Interferogram, (b) phase map, (c) unwrapped phase pattern,

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Barbosa et al. Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2877

his figure, the plate tilt angle was 30.5°, which is in ex-ellent agreement with the tilt angle set in the opticaletup ��30.3° �.

The 3D plot in Fig. 5(d) shows some statistical fluctua-ions on the reconstructed surface. These can be attrib-ted mainly to optical noise sources due to irregularitiesn the front crystal face and light scattered by the crystaldges. This noise cannot be eliminated by the PST, sincehe optical noise pattern generated by the reference beamoes not significantly change between two consecutiveteps. The fluctuations are also partially caused by thelectronic noise of the CCD camera. The absence of anybservable periodic, low-spatial-frequency deviations onhe 3D plot in Fig. 5(d) shows that the simplifications con-erning the number of laser modes considered in Section 3nd briefly discussed there are valid and provide good re-ults. The use of Eq. (6) in the PST is therefore not re-ponsible for the high-spatial-frequency, random fluctua-ions.

We tested the technique for objects with more complexeometries. Figures 6(a)–6(d) show the interferogram, thehase map, the unwrapped phase pattern, and the result-ng relief, respectively, of a cylindrical surface. The cylin-er diameter obtained by PST was 41.4±0.3 mm, which isn accordance with the value of 41.6 mm measured with a

ig. 7. The same analysis as shown in Figs. 5 and 6 for a loudsattern, (d) 3D plot.

ial caliper. The relatively large dispersion in the resultbtained by PST is mainly due to the statistical fluctua-ions mentioned above. Figures 7(a)–7(d) show the corre-ponding patterns of a (partially illuminated) loud-peaker surface.

. Two Reference Beamshe optical path difference between the reference beamsas set to be an odd multiple integer of �S eff, and a holo-raphic recording with two reference beams was per-ormed. The build-up time of the hologram was �15 s.igure 8(a) shows the interferogram of a metallic plate

ilted �22° with respect to the front face of the crystal.ote that the fringe visibility obtained is very satisfac-

ory. The PST was carried out according to Subsection 2.Dnd Eq. (10), and the displacement applied to the trans-ation stage in each step was 0.33 mm. The referenceeams must be slightly adjusted in order to correct theiruperposition at the BTO and to keep a homogeneous in-ensity distribution of the fringe pattern in each record-ng. Figures 8(b)–8(d) show the resulting phase map, thenwrapped phase pattern, and the 3D surface reconstruc-ion of the plate, respectively.

As already observed in the holographic recording withne reference beam, there are no discernible periodic de-

surface: (a) Interferogram, (b) phase map, (c) unwrapped phase

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2878 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 Barbosa et al.

iations in the fluctuations of the 3D plot shown in Fig.(d). This confirms that the assumption of three laserodes oscillating with equal intensities can be considered

alid. The most important source of the statistical fluc-uations is the optical noise generated mainly by the ref-rence beams, because they are �15 times more intensehan the object beam. Since those beams do not propagatelong the same direction, they reach different parts of therystal. Hence, the use of two reference beams makes theffect of optical noise more critical to holographic imag-ng. A possible way to overcome this limitation is to takeeveral (3 or 4) frames in each step with the referenceeams slightly readjusted between each frame to modifyhe noise pattern from one frame to another. The inter-erogram of each step is then obtained by taking the com-utational average of the frames. The optical noise gener-ted by the reference beams can also be significantlyeduced by properly adjusting the ratio between the ob-ect beam and the reference beam intensities.14 By adopt-ng one of the procedures above, the use of two referenceeams can effectively decrease the overall noise in theeasurement.

Figures 9 and 10 show the holographic images of theylindrical surface and the loudspeaker studied in subsec-ion 4.A with two reference beams. The higher spatial fre-uency of the resulting interferograms allows for a moreetailed surface visualization, leading to a more accurateualitative analysis. In single-reference-beam recordingshe interferogram spatial frequency is limited to theragg condition of the volume hologram, i.e., �� cannote set to indefinitely large values in order to get very high

ig. 8. Contour analysis of a 22°-tilted bar through holographicc) unwrapped phase pattern, (d) 3D plot.

recording with two reference beams: (a) Interferogram, (b) phase map,

ig. 9. Interferogram of the cylinder obtained by holographic re-

Fig. 10. Same as Fig. 9 for the loudspeaker surface.

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Barbosa et al. Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2879

patial frequencies; otherwise the hologram diffraction ef-ciency would progressively decrease. Since the interfero-ram resulting from two-reference-beam recording iserely a sum of two interference patterns, the maximum

nterferogram spatial frequency in this case is twice thatchievable with one reference beam.

. CONCLUSIONSe demonstrated the relief analysis of 3D surfaces

hrough holographic recording in sillenite crystals withultiwavelength, large FSR lasers. The simplifying as-

umption of diode lasers emitting three equally intenseodes allowed for simple and analytical expressions that

rovided results through the four-frame PST that are inood agreement with the real object measurements.

The fringe width and the number of oscillating modesre inversely proportional, so that holographic recordingith more than two wavelengths allows for the use ofual-reference-beam setups that consequently leads toigher-spatial-frequency interferograms. We have shownhat this recording improves qualitative surface analysis.he smaller effective synthetic wavelength leads to lessoisy measurements. However, the two-reference-beamecording tends to be more noisy than the conventionalne. The possibility of reducing the optical noise duringhe PST through the averaging of many frames in eachtep was pointed out.

In comparison to conventional profilometry techniques,he holographic methods present the advantage of nonde-tructive testing, thus allowing for the analysis of veryelicate surfaces or organic tissues. Moreover, the evalu-tion of the resulting interferogram provides qualitativend quantitative information about the whole surface un-er study. The method proposed by us is a very promisinglternative for holographic profilometry, since it presentshe additional advantage of obtaining interferograms iningle-exposure processes, which permits faster and sim-ler testing with easy-to-build optical setups.

CKNOWLEDGMENTSe are grateful to Jaime Frejlich of Universidade Es-

adual de Campinas for providing us with the BTO crystal

sed in this work. This work was partially supported byhe Fundação de Apoio à Tecnologia.

Corresponding author E. Barbosa’s e-mail address isbarbosa@fatecsp.br.

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