A novel, to our knowledge, approach to light-stripe triangulation configuration that allows for parallel,fast, real-time three-dimensional surface topography with an extremely large number of optically re-solved depth steps is presented, analyzed, and experimentally demonstrated. The method is based ona color-coding and decoding arrangement that exploits polychromatic illumination and axially dispersingoptical elements. This leads to an increase of the depth-measuring range without any decrease in theaxial or the lateral resolution. Our experiments yield three-dimensional surface measurements withlateral and depth optical resolutions of ,40 nm, for a depth of focus of 48 mm, resulting in 1.2 3 106
Optical three-dimensional ~3-D! profilometry mea-suring is essential for many fields in science andindustry, observation of micro-optoelectromechanicalsystems, machined parts, integrated circuits, andbiologic specimens.1–3 There are a variety of electro-optical measuring approaches available for three-dimensional analysis of structures in the macroscopicrange as well as in the microscopic. However, thestructured light-triangulation method is the mostwidespread for shape measurement of a diffusingsurface.4 It is suitable for scientific and industrialapplications in that it offers a simple and robust 3-Dmeasurement. The simplest structured light sys-tem projects a single point of light from the sourceonto an object. The point is then imaged with off-axis configuration, on a lateral effect photodiodes orlinear arrays of detectors. The single point triangu-lation approach is relatively inexpensive and has ahigh resolution. However, measuring the surface of3-D objects involves lengthy time-consuming scan-ning, which is often impractical. A more compli-cated structured light system operates by projecting a
The authors are with the Optical Engineering Laboratory, Fac-ulty of Mechanical Engineering, Technion-Israel Institute of Tech-nology, Haifa 32000, Israel. E. Hasman’s e-mail address [email protected].
Received 30 March 2000; revised manuscript received 2 January2001.
light stripe onto the object and using a two-dimensional detector array for simultaneous mea-surement of a linear cross section.5 Fewer framesare then required for measuring the entire 3-D object,and scanning need be done only in the direction per-pendicular to the stripe.
Unfortunately, the conventional structured light-triangulation approaches, described above, cannot si-multaneously achieve a large depth measuring rangeand a high lateral resolution. This is because theconventional optical lenses that are incorporated inthese systems cannot, at the same time, provide bothlong focal depth and high lateral resolution. Specif-ically, high lateral resolution requires high numericalapertures, whereas large depth of focus requires lownumerical apertures. The relation between the lat-eral resolution 1yDx ~Dx is the spot size! and thedepth of focus dF is given by
dF 5 k~Dx!2yl0, (1)
where l0 is the wavelength of the light and k is aonstant number between 1 and 6 depending on thexact definitions of Dx and dF and on the wave-frontpodization.A common trade-off between the lateral resolution
nd the depth of focus is to reduce the lateral reso-ution, i.e., to use a relatively large spot, the center ofhich can be determined with much higher accuracy
han the spot size, by complicated numerical tech-iques.6 Such techniques can overcome, somewhat,
the errors that are due to the statistical noises, suchas shot noise, CCD amplifier noise, CCD pixel re-
1 April 2001 y Vol. 40, No. 10 y APPLIED OPTICS 1609
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sponse nonuniformity, and quantization noise. Un-fortunately, this approach is highly sensitive tosystematic noises such as local changes in the reflec-tivity or the shape of the object ~e.g., short radius ofcurvatures cannot be adequately dealt with!. More-over, for coherent illumination, speckle noise severelylimits the possibility of achieving subspot sizeresolution.7–9 An alternative approach, applicableo the single-point triangulation approach, is to usespherical optical elements with a high resolutionnd long focal depth such as Axicon10,11 or Axilens.12
Unfortunately, these approaches cannot be directlyadopted to the light-stripe configuration. Specifi-cally, using such aspherical optical elements to forma light stripe with high lateral resolution and longfocal depth will result in high side lobes; conse-quently, the peak-to-background ratio is relativelylow.13
Recently, we presented what to our knowledge is anovel approach for extending the focal depth of light-stripe triangulation, using the color-coded approachin which these limitations were largely reduced andgood peak-to-background ratios were obtained.14
Although 18-fold improvement of the focal depth wasdemonstrated, the overall depth resolution was lim-ited by detector noise and lack of sophisticated imageprocessing to .10 mm, and the number of resolving
epth steps ~NRD! was a few thousand.In this paper we present a comprehensive theoret-
cal and experimental investigation of 3-D opticalrofilometry with light-stripe triangulation based onolor coding and decoding that permits parallel, fast,eal-time 3-D surface mapping with a large deptheasuring range as well as high axial and lateral
esolutions. The method is based on a color-codedrrangement that exploits polychromatic illumina-ion and a cylindrical element that axially ~longitu-inally! disperses the incident light in order toncrease the depth measuring range without any de-rease in the vertical and horizontal resolution. Wese an on-axis cylindrical diffractive optical elementDOE! and a combined diffractive–refractive opticallement, whereby many light stripes, each of a dif-erent wavelength, are simultaneously focused at dif-erent focal lengths, forming a rainbow light sheet.
We investigate two main configurations. In therst configuration the rainbow light sheet is usedirectly to obtain the surface profiles. Here the in-rease in the focal depth is accompanied by a reduc-ion in the peak-to-background ratio ~although at auch smaller amount than the monochromatic con-
gurations!. In the second approach, color decodings added by means of a variable-wavelength filterVWF! whose dispersion is exactly matched to that ofhe rainbow light sheet. Both our theoretical anal-sis and our experimental results reveal that, withptimal decoding, the diffraction-limited spot sizend shape are completely maintained even for a largencrease in the depth of focus. Finally, low noiseetection and sophisticated digital signal postpro-essing are used to obtain ,40-nm depth resolution,
610 APPLIED OPTICS y Vol. 40, No. 10 y 1 April 2001
ielding .106 of resolving depth steps over the entirefocal depth.
2. Basic Principle of the Measuring System
The operation of our color-coded light-stripe triangu-lation system is described with the aid of Fig. 1. Thebroadband light source ~multicolor! can be either a
hite-light source or a short-pulsed laser that pro-uces a relatively large spectral bandwidth. A col-imating lens system forms plane waves that areocused by axially dispersing optics ~ADO! such as aylindrical DOE or a hybridycombined diffractive–efractive optical element. The ADO forms a rain-ow light sheet that consists of light stripes ofifferent wavelengths ~color! at different distancesrom the lens. An object with maximum height dif-erence smaller than the rainbow focal depth DF islaced in the region of the rainbow light sheet. Thebject intersects the rainbow light sheet, and the in-ersection profile of the object is then imaged with anff-axis configuration ~at an angle u from the illumi-ation optical axis! to a two-dimensional CCD cam-ra. Although the rainbow light sheet is composedf thin stripes ~limited by the diffraction! of the indi-idually focused wavelengths, each stripe is sur-ounded by background light of other wavelengths.hus the detected profile is relatively broad. Thisrofile can be significantly narrowed by means ofuppressing the background light and allowing theight of only the proper ~focused! wavelength to beetected at each object depth. This is achieved bynsertion of an interference filter into the detectionath whose transmitted wavelength is spatially non-niform. With such a VWF, optimal performance is
obtained when the spread and specific wavelengthlocation matches that of the light from the axiallydispersing optics. When the match is exact, it ispossible to obtain a diffraction-limited resolution overthe entire depth of focus without any scanning. Thedata that are detected by the CCD camera are dis-played on a monitor and processed by the computer togive the intersection profile in virtually real time.After a line is detected, a computer-controlled steppermotor shifts the object to get another line, and so on,until a complete 3-D profile of the object is obtained.
We can evaluate the color-coded triangulation ap-proach by resorting to a simple geometrical analysisand to physical optics considerations. The axiallydispersing optics can be an on-axis cylindrical DOE ora hybrid combination of diffractive–refractive lens.Figure 2 illustrates a hybrid lens that is illuminatedwith a broadband light beam. The hybrid lens in-cludes a plano–convex cylindrical lens with a diffrac-tive element that has blazed grooves ~kinoform! onthe planar surface. The hybrid lens serves to focuson light stripes of different wavelengths at differentdistances from the lens. For example, the wave-length l1 is focused to the focal length of F1, l0 isfocused to F0, and l2 is focused to F2. For an ideal-ized one-dimensional quadratic cylindrical DOE, thetransmission function t~x! is given by
t~x! 5 exp@if~ x!# 5 expS2ipx2
l0f0D , (2)
here x is the lateral coordinate in the thin DOE, fs the phase function, and f0 is the focal length of the
DOE for a wavelength l0. When such DOE is illu-minated with another wavelength l, the resultingfocal length fd~l! is given by
Equation ~3! indicates that the focal length is in-versely proportional to the wavelength, which is theaxial dispersion of the quadratic DOE. By use of aDOE with the quadratic phase function, it is possibleto obtain the range of the rainbow light sheet ~depthf measuring range! by the relation
DF 5 Dz 5 f0~Dlylo!, (4)
where Dl 5 l2 2 l1 is the illuminated wavelengthband ~l2 and l1 are the upper and the lower wave-lengths, respectively! and l0 5 ~l2l1!1y2. To matchthe dispersion of the ADO to a commercially availableVWF or to change the depth measuring range DFwhile keeping the same effective focal length, it maybe necessary to combine diffractive and refractiveelements. In such a hybrid combination, the refrac-tive lens has a focal length fr~l! and the DOE has aocal length fd~l! yielding an effective focal length
F~l!. A simple expression for the focal depth of thehybrid element can be approximated, assuming neg-ligible separation between the refractive and the dif-fractive elements, and neglecting the dispersion ofthe refractive lens, as follows,
DF > SF0
f0DSF0
Dl
l0D , (5)
where l0 5 ~l2l1!1y2 @assuming l0 > ~l2 1 l1!y2#, F05 F~l 5 l0! and f0 5 fd~l 5 l0!. The enlarging factorof the focal depth Mo is defined as the ratio betweenthe extended focal depth of the hybrid diffractive–refractive element, given by relation ~5! and themonochromatic focal depth, dF 5 4l0F#
2 ~80% of themaximal intensity!, where F# is the f-number of thehybrid lens. With this definition we get
Mo 5DFdF
> SDl
l0DS D2
4l0f0D 5 SDl
l0D NDOE , (6)
where D is the aperture diameter of the ADO andNDOE is the Fresnel number of the diffractive ele-ment. NDOE is also the axial resolving power of theADO, i.e.,
NDOE >l0
dlDL, (7)
where dlDL is the wavelength range that maintains adiffraction-limited spot. Equation ~6! indicates thathe enlarging factor of the focal depth depends onlyn the Fresnel number of the DOE and on the illu-inated spectrum band.To obtain high axial and lateral resolutions in theeasuring process, the rainbow light sheet is ob-
erved through a VWF, whereby the wavelength-ransmittance function along the length of the filterxactly matches the axial dispersion of the ADO. Inddition to completely restoring the diffraction-imited spot size, we should set a the filter transmit-ance bandwidth dlf equal to ~or smaller than! dlDL of
relation ~7!. Defining the VWF discrimination Mf as
1 April 2001 y Vol. 40, No. 10 y APPLIED OPTICS 1611
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the number of the wavelength that can be discrimi-nated, i.e.,
Mf 5Dl
dlf, (8)
and using relations ~6! and ~7! yields the expectedoptimal filter discrimination as Mf
opt > Mo. As wehow below, using VWF with such optimal Mf indeed
yields a sharp image of the object profile withdiffraction-limited and low sidebands. A commer-cially available VWF is a linearly variable interfer-ence filter, so it is desirable to obtain ADO with linearaxially dispersion. This can be obtained with excel-lent accuracy in the limit f0 .. fr.
From geometrical consideration, the resolvabledepth of the system is given by
dz 5dximg
sin u, (9)
here dximg is the resolvable lateral shift of the im-ged light stripe ~the center! on the CCD. In gen-ral, the resolvable lateral shift of the light stripe isroportional to the imaged stripewidth itself, wherehe proportional constant depends on the accuracyeeded to determine the center of the image lightistribution. A commonly used merit function toharacterize the performance of the color-coded opti-al profilometer is the NRD as
NRD 5DFdz
5 Mo
dFdz
. (10)
Equation ~10! indicates that, for our color-coded sys-tem, the NRD can be increased by the factor of Mo,compared with the conventional monochromaticlight-stripe triangulation, while maintaining thesame lateral resolution.
3. Realization and Experimental Results
An automated 3-D optical profilometer was designed,realized, and evaluated experimentally and theoret-ically for the acquisition and measurement of 3-Dsurfaces. For the illumination source we used acontinuous-spectrum 75-W xenon arc lamp of spec-trum l 5 0.4–0.7 mm and used a heat-absorbinglass to reduce the IR radiation. The light emergingrom the lamp is focused by a parabolic mirror onto alit with a 15-mm diameter. The slit is imaged with4-f system. The first lens in the 4-f system is an
chromatic lens with 360-mm focal length, and theecond lens is the ADO, a combination of a cylindricalefractive lens designed to have fr 5 496 mm at l0 5
0.529 mm and a quadratic diffractive element,hereas the phase function is given by Eq. ~2! and f0
5 fd~l0 5 0.529 mm! 5 1040 mm. The gap betweenthe cylindrical refractive lens and the diffractive el-ement is 17 mm, owing to practical constraints. Theaperture diameter of the combined lens is D 5 9 mm.The diffractive element was realized with photolitho-graphic techniques and reactive ion etching to pro-
612 APPLIED OPTICS y Vol. 40, No. 10 y 1 April 2001
duce a 16-binary-level element on a fused-silicasubstrate.15
The measured and the calculated dispersion re-sults for the cylindrical refractive lens and DOE sep-arately and combined are shown in Fig. 3. For themeasurements we used the white-light source andthe VWF for selecting the desired wavelength. Thecylindrical refractive lens was a plano–convex cylin-drical lens made of BK7 material. The predicteddispersion of the cylindrical refractive lens is ob-tained from the dependence of its index of refractionon the wavelength, given by
n~l! 5b1l
2
l2 2 c11
b2l2
l2 2 c21
b3l2
l2 2 c3,
where b1 5 1.0396, b2 5 0.23179, b3 5 1.01047, c1 56.0007 3 10–3, c2 5 2.00179 3 10–2, and c3 5 1.0356 3102, and the wavelength is given in micrometers.Using the thin-lens approximation, the refractivelens dispersion is given by
fr~l! 5fr~l0!@n~l0! 2 1#
n~l! 2 1.
or the DOE the theoretical prediction for the dis-ersion, fd~l!, is given by Eq. ~3!. With the thin-lens
approximation the theoretical dispersion relation ofthe combined diffractive–refractive lens with a gap dbetween them is given by
F~l! 5fd~l!@ fr~l! 2 d#
fr~l! 1 fd~l! 2 d.
s evident, there is excellent agreement between theheoretical prediction and the experimental measure-ents of the dispersions of the refractive, diffractive,
nd combined optical elements ~ADO!. The disper-ion of the ADO was measured to have a total focal
Fig. 3. Measured and calculated dispersion of the cylindricallenses ~l0 5 529 nm!; refractive lens, measured ~crosses! and cal-culated ~dotted–dashed curve!; diffractive lens, measured ~pluses!and calculated ~dashed curve!; combined diffractive–refractivelens, measured ~circles! and calculated ~solid curve!.
c
sFa
>
e
s
c
e
length F0 5 326 mm at l0, the focal depth of thiscombined lens was DF 5 48 mm, whereas the mono-hromatic focal depth ~conventional! for F# 5 36 was
dF > 2.7 mm. Therefore the enlarging factor of thefocal depth given by Eq. ~6! is Mo > 18, which corre-ponds approximately to the predicted Mo using theresnel number of the diffractive optics as NDOE > 36nd Dlyl0 > 0.567. The dispersion of our ADO was
designed to be approximately linear ~shown in Fig. 3!;therefore the VWF in our experiments was the linearvariable interference filter, SCHOTT VERIL S 60.The transmittance wavelength of the VWF variesalong the 42-mm length of the filter, whereas thespectral width dlf > 15 nm, and the spectral band Dl
300 nm, yielding the VWF discrimination to be Mf> 20 @Eq. ~8!#, corresponding to the optimal consid-ration, Mf
opt > Mo. The measured dispersion of thecombined diffractive–refractive lens fits inside thelinear VWF transmittance.14
Fig. 4. Measured ~crosses! and calculated ~solid curve! diffractionfficiency of the DOE as a function of the wavelength.
Fig. 5. Measured sections of the imaged light-intensity distributio~a! without VWF and ~b! with VWF.
The first-order diffraction efficiency for a multilevelbinary DOE depends on the illumination wavelengthl. When we combine the results of Refs. 15 and16,the diffraction efficiency h1 as a function of l can behown to be
h1 5 3sinSp
NDp
N4
2
3 sinSpld 2 l
l DN sinSp
Nld 2 l
l D42
, (11)
where N is the number of level and ld is the designedwavelength. In our experiments N 5 16 levels. Wehoose ld 5 0.509 mm so that the lowest diffraction
efficiency in the visible spectrum ~0.4–0.7 mm! ismaximal. The measured and the theoretical predic-tions of the diffraction efficiency as a function of thewavelength are given in Fig. 4. As can be seen thereis a good agreement between the predicted and themeasured efficiencies, where .75% diffraction effi-ciency was obtained for the entire visible spectrum.
Two main configurations of the color-coded pro-filometer were investigated experimentally and the-oretically. In the first configuration we used therainbow light sheet directly without inserting theVWF, whereas in the latter a color-decoding tech-nique was used with the VWF. Figure 5 shows mea-sured sections of the imaged light-intensitydistributions of the intersection between the rainbowlight sheet and the flat object ~a! without VWF and ~b!with VWF. The improvement with the VWF isclearly evident. The intensity distribution withoutcolor decoding is significantly broad compared withthe color-coding and decoding approach.
To test the experimental arrangement of the twoconfigurations more quantitatively, we performed aseries of measurements on a flat object, placed at anangle so as to include the entire focal range in themeasurements. First, we measured the intensity
the intersection between the rainbow light sheet and a flat object:
ns of
1 April 2001 y Vol. 40, No. 10 y APPLIED OPTICS 1613
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cross-section distributions of the reflected light atthree different locations along the focal range at thebeginning of the focal range, z 5 299 mm; the middleof the focal range, z 5 323 mm; and at the end of thefocal range, z 5 347 mm. These measured intensitydistributions were then compared with calculated in-tensity distributions. The predicted monochromaticintensity distributions were calculated by use of theFresnel diffraction integral, given by
I~x0, z , l! 51
~lz!2 U*exp@ifL~ x1, l!#
3 expFip
lz~ x0 2 x1!
2G dx1U 2 , (12)
where the quadratic phase of the lens is fL~x1, l! 52$~px1
2!y@lF~l!#%, with the total focal length of theombined diffractive–refractive elements F~l!. Theredicted polychromatic intensity distributions arealculated by
Ip~xo, z! 5 *S~l!RVWF~l!I~ x0, z, l!dl , (13)
here S~l! is the light-source spectrum and is thepectrum response of the VWF centered at theavelength, which is focused at the calculated dis-
ance ~focal length! z @note that RVWF~l! 5 1, foroptical arrangement without the VWF#. The mea-ured and the calculated results are presented inigs. 6 and 7. Figures 6~a! and 6~b! show the mea-ured and the calculated intensity cross sections,espectively, without use of the VWF. Both theeasurement and the calculation reveal a broadndesirable background illumination combinedith a small and narrow peak. Surprisingly, the
alculations predict that the narrow peak remainslearly detectable above the broad background evenhen the enlarging factor of the focal depth, Mo, is
considerably large, ~e.g., Mo . 500!. This by itselfepresents a plane version of a nondiffracting beam,nalogs to the circular Bessel beam17 produced by
Axicon.11,18 Figures 7~a!–7~c! show the measurednd the calculated intensity cross sections when aWF is used. As evident, there is a good agree-ent between the calculated and the measured in-
ensity cross sections. The narrowing of thentensity cross sections, because of the VWF, islearly seen, indicating that near-perfect matchingetween the ADO and filter dispersions was indeedbtained. Moreover, these results show that aiffraction-limited linewidth of approximately Dx >0 mm at FWHM is maintained throughout the fo-al range when the VWF is used.The resolvable depth of the system given by Eq.
9! depends on the accuracy required for determin-ng the center of the image light distribution.herefore an important quality criterion is the ratioetween the maximal value of the peak to that ofhe sidelobe value, also known as the peak-to-ackground ratio. The peak-to-background ratiorovides a limit on the allowed noise level that cane introduced into the system. Higher noise levels
614 APPLIED OPTICS y Vol. 40, No. 10 y 1 April 2001
ignificantly degrade the detected spot-size. Forxample, the results of Fig. 6 indicate that the peak-o-background ratio without a VWF is only ;1.5.n this case the noise level should be limited to0% of the maximal background value. WithWF, however, the peak-to-background ratio im-roves significantly to ;15, and the permitted noisean reach values of 1500% of the maximal back-round value without significantly degrading its ac-uracy.To determine the optical resolvable depth of our
ystem with the VWF, it is necessary to scan an objectith a known surface topography and compare theeasured results with the given surface. This was
one with a flat object ~a high-quality laser mirror!laced at an angle. The measured profile of the flatbject is shown in Fig. 8. For this measurement wesed a 12-bit digital-cooled CCD camera, 1280 3024 pixels ~Sensicam, PCO Computer Optics GmbH!nd the centroid ~center of gravity! algorithm to de-ect the center of the imaged line. The rms deviationf the measured result from the expected linear lineas ,40 nm. These results indicate that, for our
Fig. 6. Intensity cross sections at three positions along the focalrange, without the VWF: ~a! measured and ~b! calculated.
ashed curve, z 5 299 mm; solid curve, z 5 323 mm; dotted–ashed curve, z 5 347 mm.
ca~
laioodct
cm
lateral and depth resolutions of dx > dz > 40 nm ~u 5py2!, and for the extended depth measuring range ofDF > 48 mm, the number of NRD > 1.2 3 106 @ac-ording to Eq. ~10!# can be obtained. Note that, with
conventional incoherent triangulation approachwithout color coding and decoding! and comparable
Fig. 7. Measured ~dots! and calculated ~solid curves! intensityross sections at three positions along the focal range for ~a! z 5 299m, ~b! z 5 323 mm, ~c! z 5 347 mm.
noise level, the NRD would be only dFydz ' 6 3 104.Also note that in conventional coherent ~laser! trian-gulation systems the axial and the lateral resolutionsare limited by speckle noise,7–9 which often stronglyreduces the NRD. However, in our approach withincoherent light, ;500-fold superresolution was ob-tained and, combined with an 18-fold increase of thefocal range, yielded an extremely high NRD.
Our profilometer used real-time software for read-ing the data from the CCD camera and processing thecenter calculations. A single line cross section of theobject’s profile is obtained at video rates, so the timeto complete a full 3-D scan of the object’s surface wasdetermined by the speed of the stepper motor.
4. Conclusions
We have presented what to our knowledge is a novelapproach for rapidly determining the surface of 3-Dobjects. It is based on a color-coding arrangementthat exploits polychromatic illumination and cylin-drical, axially dispersing optics to increase the depthmeasuring range without any decrease in the lateraland axial resolutions. In a related approach a VWF,spectrally matched to the disperse illumination, isadded to the observation system for color-decoding, toresolve completely the diffraction-limited spot sizeand shape through the entire depth measuring range.Our approach is also valid for even larger increase ofthe depth measuring range ~e.g., Mo . 500! and forenses with a small f-number, but more elaborateberration-correction design for the axially dispers-ng optics will be needed. The main advantage ofur approach over the conventional triangulationnes is that it exploits the additional degrees of free-om provided by the multiwavelength illumination inonjunction with the spatial data. This is in con-rast to previous color-coding schemes19 that use only
the wavelength information instead of the spatialinformation, resulting in relatively poor discrimina-tion ability.
Fig. 8. Section of a measured profile of a flat object ~pluses!. Alsoshown is the expected linear object ~dashed curve!.
1 April 2001 y Vol. 40, No. 10 y APPLIED OPTICS 1615
10. G. Bickel, G. Hausler, and M. Maul, “Triangulation with ex-
1
This research was supported by the fund for thepromotion of research at the Technion-Israel Insti-tute of Technology.
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