Simple multifrequency and phase-shiftingfringe-projection system based ontwo-wavelength lateral shearing interferometryfor three-dimensional profilometry
Dalip Singh Mehta, Satish Kumar Dubey, M. Mosarraf Hossain, and Chandra Shakher
In recent years various optical noncontact techniquesfor measuring three-dimensional (3-D) shapes of dis-continuous objects have been developed.1,2 Projectinga sinusoidal fringe pattern,3 a grating4 or structuredlight,5 or a moiré pattern6 onto the objects has beenthe most widely used technique for 3-D shape mea-surement of discontinuous objects. The common prin-ciple of all these techniques is that a periodic patternis projected upon an object and the deformed patternfrom the object is observed from a different view an-gle. The deformed patterns are then analyzed byphase-shifting or Fourier-transform methods, alsocalled phase-shifting profilometry7 (PSP) or Fouriertransform profilometry (FTP),8,9 respectively, and the3-D shapes of the objects are reconstructed. Thesetechniques have proved to be highly accurate and
efficient and can be used in a robust environment.One of the key devices used in these techniques is theefficient generation of periodic patterns with changesin the spatial-carrier frequency and also with phase-shifting patterns in real time. So far most fringe orstructured light-projection equipment has been basedon interferometric systems10; liquid-crystal displayprojectors11,12; digital micromirror devices13; andsingle-, dual-, and multiple-frequency gratings.14–16
Fringe-projection systems based on Michelson-typeinterferometric configurations10 are highly sensitiveto external vibrations and therefore cannot be usedin robust environments. Fringe-projection systemsbased on liquid-crystal display projectors and digitalmicromirror devices are programmable and computercontrolled and hence have several advantages, suchas the ability to introduce phase shifting, changes inspatial-carrier frequency, and frequency multiplex-ing of fringes. But these systems are extremely ex-pensive and hence are not affordable. Production ofhigh-quality dual- or multiple-frequency gratingsalso requires special techniques. Further, once thegratings are fabricated one cannot change thespatial-carrier frequency of the grating in real time,and these systems are also expensive.
The authors are with the Laser Applications and HolographyLaboratory, Instrument Design Development Centre, IndianInstitute of Technology Delhi, Hauz Khas, New Delhi 110016,India. D. S. Mehta’s e-mail address is [email protected].
Lateral shearing interferometry (LSI) has been apotentially powerful technique for testing laser beamcollimation and optical components and for the mea-surement of the refractive-index distribution oftransparent objects, fluid flow, diffusion, vibration,and temperature.17–22 There are several optical ar-rangements for constructing LSI, among which asingle-element lateral shearing interferometer, suchas a parallel plate or a slightly wedged plate, is mostfrequently used.17 Recently various phase-shifting tech-niques were also implemented in single- or two-elementLSI.23–26 In single-element phase-shifting shearing inter-ferometers23,24 the phase shifting can be achieved eitherby application of voltage to a liquid crystal sandwichedbetween two glass plates23 or by the movement of awedge-shaped shear plate.24 In two-element LSI, how-ever, one can achieve phase shifting either by moving oneof the wedge plates25 or by using a piezoelectric trans-ducer phase shifter.26 With the incorporation of phase-shifting techniques,23–26 shearing interferometry hasbecome an increasingly important field of research anddevelopment. There has also been a large amount of re-search and development in the field of two-wavelengthinterferometry27–37 to extend the range of unambiguity inthe measurement of distance and of the step height ofdiscontinuous objects. Most previous studies of two-wavelength interferometry have used Michelson-type orunbalanced interferometric systems. These interferomet-ric systems are highly unstable, difficult to align, requirehigh-quality multiple optical components (mirrors, beamsplitters, etc.), and are expensive and prone to externalvibrations.
In this paper we report the development of a mul-tifrequency spatial-carrier fringe projection system
based on two-wavelength LSI.38 From the same setupa variety of sinusoidal fringe patterns were gener-ated; i.e., we obtained changes in the spatial-carrierfrequency of the fringes either by changing the wave-length of the laser light or by slight defocusing, andwe obtained the phase-shifted fringe patterns by sim-ply moving the wedge plate in an in-plane paralleldirection, using a linear translator. A synthetic inter-ferogram of slow spatial-carrier frequency was alsoobtained by simultaneously switching on both of thelasers. One of the main advantages of the multifre-quency fringe-projection system is that it provides anappropriate and more-accurate phase-unwrappingmechanism for objects with large discontinuities.14,16
In these methods, multiple images are recorded atfringe patterns with different periods or frequencies,and the phase at each pixel is unwrapped. The fringe-projection system that we developed was tested formeasurement of the three-dimensional shape of adiscontinuous object. Compared with other tech-niques for generating fringes, LSI has many advan-tages: It is a common-path interferometry and henceis insensitive to external vibrations, is compact insize, and is relatively inexpensive. Because of theseadvantages this technique is most suitable for indus-trial applications in robust environments.
2. Principle of the Fringe-Projection System Based onTwo-Wavelength Lateral Shearing Interferometry
A. Generation of Multifrequency Spatial-Carrier Fringes
Figure 1 is a schematic diagram of the common-pathtwo-wavelength lateral shearing interferometer. Agreen diode laser with a wavelength of �1 � 532 nm
Fig. 1. Schematic diagram illustrating a two-wavelength lateral shearing interferometer based on a wedged plate.
and a red He–Ne laser with a wavelength of �2� 632.8 nm, each with 5 mW of power, were mixed atbeam splitter BS. The calculated synthetic wave-length, � � �1�2���2 � �1�, is 3.34 �m, and averagewavelength � � �1�2���1 � �2� is 0.289 �m. Bothbeams were then passed through a beam expanderand a spatial filter unit simultaneously. A collimatinglens of 250 mm focal length and 50 mm diameter wasused to collimate both beams. The collimated beamswere then made incident onto a lateral shearing in-terferometer made from a slightly wedged plate of5 mm thickness. Interference fringes are formed be-tween the two laterally shifted wavefronts in the re-gion of superposition for each wavelength withdifferent spatial-carrier frequencies. The wavefrontunder test for �1 and �2 may be expressed as W�x, y�,where �x, y� are the coordinates of point P. Whenwavefront W�x, y� is sheared in the x direction by anamount S, the sheared wavefront at the same point isW�x � S, y�. The resultant optical path difference(OPD), �W�x, y�, between the original and thesheared wavefronts can be written as17
�W�x, y� � W�x, y� � W�x � S, y� � Ek y � nk�k, (1)
where Ek, nk, and �k (with k � 1, 2) are tilt angle,order of interference fringes, and wavelength, respec-tively. Subscript �k � 1, 2� is used throughout thispaper to differentiate between �1 and �2. If defocusingand tilt are simultaneously present, the OPDs forboth beams are given by
where Dxk �k � 1, 2� is the defocusing. Equation (2)represents a system of straight fringes that are par-allel neither to the x axis nor to the y axis, i.e., areinclined in the x direction with slope k �k � 1, 2�given by
k � tan�1�2DxkSEk
�. (3)
From Eq. (2) it can be noted that in LSI based on aparallel plate without tilt �Ek � 0� a fringe-free fieldwill be generated at focus �Dxk � 0�, and therefore nospatial-carrier frequency will be generated. For LSIbased on a slightly wedged plate, however, even atfocus �Dxk � 0� a finite number of interference fringesare produced because of the presence of tilt, whichgenerates a significant spatial-carrier frequency par-allel to the direction of shear. Therefore one can usethe Fourier-transform method8,9 for fringe analysis inLSI based on a wedge plate. This is so because for theFourier-transform method for fringe analysis it isnecessary that a large-spatial-carrier frequency beintroduced by tilting one of the wavefronts such thatthe first-order spectrum is sufficiently separated from
the zero-order spectrum and can easily be filteredout. In the presence of two beams with differentwavelengths �1 and �2 simultaneously propagating inthe same optical path (see Fig. 1), the beat formationphenomenon can take place, and a third interfero-gram corresponding to a synthetic wavelength can beobserved. Therefore a third interference fringe pat-tern for projecting on the discontinuous object is gen-erated with a different spatial-carrier frequency, andthe OPD for a synthetic ��� wavelength can be writ-ten as
�W�x, y� � W�x, y� � W�x � S, y� � Ey � N�, (4)
where N is the integer synthetic order of interferenceand W�x, y� is the OPD at a synthetic wavelength.When S is small, the OPDs can be written as17
�Wx �S � nk�k, (5)
�WX �S � N�. (6)
Thus there are three interference fringe patterns thatcorrespond to �1, �2, and � with three spatial-carrierfrequencies. The phases �1�x, y�, and �2�x, y� of theinterference fringes at individual wavelengths �1 and�2 can be expressed, respectively, as
�1�x, y� �2�
�1OPD, (7)
�2�x, y� �2�
�2OPD. (8)
On subtracting Eq. (8) from Eq. (7) we obtain phase �x, y� at a synthetic wavelength:
�x, y� � �1�x, y� � �2�x, y�
� 2�� 1�1
�1�2�OPD �
2�
�OPD. (9)
Another advantage of LSI is that, as the object ismoved outside and inside the focus, the orientation ofthe fringes and spatial-carrier frequency also chang-es.39 Therefore fringes at different angles can also beprojected at multiple frequencies onto the objects.
B. Generation of Phase-Shifted Fringe Patterns
Various phase-shifting methods have been implementedin single- or two-element LSI.23–26 One of the simplestphase-shifting methods employed in single-element LSIis the movement of a wedge-shaped shear plate in aplane-parallel direction by use of a linear translator.24 Weemployed a similar method to generate the phase-shiftedinterferograms for projecting upon the discontinuous ob-jects. In wedge-plate LSI, the relation between the mea-
sured OPD and lateral shear S is given by Eq. (1). Lateralshear S can be expressed as17,24
S �h sin 2�
�n2 � sin2 ��1�2, (10)
where h is the plate’s thickness, � is the angle ofincidence, and n is the refractive index of the plate.From Eq. (10) it can be seen that lateral shear S is afunction of the plate’s thickness, and in a wedge platethe thickness is variable along the wedge direction.Therefore, moving the wedge plate in an in-plane-parallel direction would introduce an additionalOPD24 and would produce the required phase shift inphase-shifting interferometry (PSI). The amount ofphase shift changed by movement of the wedge can beexpressed by24
�k��0� �2�
�kOPD�
2�
�k��0 cos �
�4���0�nk
2 � sin2 ��1�2
�k, (11)
where �0, �, and � are the distance to be moved, theangle of deviation, and the wedge angle, respectively,and nk �k � 1, 2� is the refractive index of the wedgeplate at wavelength �k. The necessary moving dis-tance required for 2� phase shift in PSI is
�2� ��k
2��nk2 � sin2 ��1�2. (12)
The four phase-shifted shearing interferograms de-rived at a wavelength of 633 nm by the techniquedescribed above were discussed in Ref. 24. FromEq. (12) it is clear that, by changing the wavelength,one can also obtain phase-shifted interferogramswith different spatial-carrier frequencies. Thus withthe present techniques a variety of fringe patternscan be obtained for 3-D surface profilometry.
3. Experimental Results
A schematic of the experimental setup of two-wavelength LSI is shown in Fig. 1 and was described inSubsection 2.A. First, the sheared interferograms wereprojected onto a reference plane; then they were re-corded by a charge-coupled device (CCD) detector(Roper Scientific, Inc.) that had 1040 � 1392 pixels,with each pixel size being 6.5 �m � 6.5 �m. Inter-ferograms were recorded first when the collimatinglens was placed exactly at focus position from thespatial filter and the beam was perfectly collimated.During the recording the following procedure wasadopted: First, both laser beams were switched onsimultaneously and the interferogram was recorded;then interferograms that corresponded to individualwavelengths �1, and �2 were recorded by switchingon one of the lasers and closing the other and vice
versa. Figures 2(a), 2(b), and 2(c) show the shearinginterferograms that correspond to �1, �2, and �, re-spectively, at focus. As mentioned above, because ofthe presence of tilt (wedge plate), a finite number offringes were obtained, even at the focus position, andhence the Fourier-transform method can be used forfringe analysis. It can also be seen from Figs. 2(a)–2(c) that the fringes are parallel to the direction ofshear and that the spatial-carrier frequency of thefringes is different for the interferograms that corre-spond to �1, �2, and �. A two-dimensional fast Fouriertransform of each interferogram was then computed,which gave three Fourier spectra, i.e., 0-order and�1st-order spectra. Figures 2(d), 2(e), and 2(f) are theFourier spectra of interferograms that correspond to�1, �2, and �, respectively. At focus, as the spatial-carrier frequency is in the x direction, the Fourierspectra are also parallel to the x direction. It is alsoobservable from the Fourier spectra that the spatial-carrier frequency is different for all three spectra andthat the first-order spectra are sufficiently separatedfrom the zero-order spectrum. Figure 2(f) is an inter-ferogram that corresponds to synthetic wavelength�. It can be seen from this interferogram that thebeat frequency is very slow; this is so because theinterferogram is recorded at the focus position andbecause the tilt angle between the two wavefronts isvery small. Under these circumstances the first-orderspectrum of the beat frequency will not be widelyseparated from the zero order and hence cannot beused for FTP and is suitable only for PSP. The inter-ferograms that correspond to �1 and�2, however, i.e.,Figs. 2(a) and 2(b), respectively, are suitable for bothFTP and PSP.
The spatial-carrier frequency of the fringes wasfurther increased by defocusing. We achieved defo-cusing by moving the collimating lens inside or out-side the focus.39 The interferograms were recorded inthe same manner as described above. Figures 3(a),3(b), and 3(c) show the shearing interferograms thatcorrespond to �1, �2, and � at a defocused position.Another advantage of wedge-plate LSI is that, as the
collimating lens is moved outside and inside the fo-cus, the orientation of the fringes changes in theclockwise and counterclockwise directions, respec-tively,39 as can be seen from Figs. 3(a)–3(c): The in-terferograms are oriented clockwise, and the spatial-carrier frequency is remarkably increased for allthree interferograms, including the beat frequency.One can further increase the spatial-carrier fre-quency by simply changing the amount of defocusing.It can also be seen from Figs. 3(a)–3(c) that thespatial-carrier frequency of the fringes is different forinterferograms that correspond to �1, �2, and �. Un-der these circumstances, all three fringe patternsthat correspond to �1, �2, and � are suitable for bothFTP and PSP. A two-dimensional Fast Fourier trans-form of each interferogram was also computed in thiscase, and Figs. 3(d), 3(e), and 3(f) are the Fourierspectra of interferograms that correspond to�1, �2, and �, respectively. The spatial-carrier fre-quency enhancement can be clearly seen from theFourier spectra [Figs. 3(d)–3(f)] compared withFigs. 2(d)–2(f). Further, as expected, the spatial-carrier frequency is also different for all three spec-tra, and the first-order spectra are sufficientlyseparated from the zero-order spectrum. Figure 3(c)is the interferogram that corresponds to syntheticwavelength �. It can be seen from this interferogramthat beat frequency is increased remarkably andhence is suitable for both FTP and PSP. Therefore itcan be noted that by this method one can obtain thedesired spatial-carrier frequency of fringes.
The fringe-projection system described here wasalso tested for generating phase-shifted interfero-grams. First the collimating lens was moved at thefocus position such that the parallel fringes could beobtained, and one of the laser sources, i.e., the redlaser, was switched on, with the green laser kept off.We achieved phase shift by simply moving a wedge-shaped shear plate in an in-plane parallel direction,using a linear translator as shown in Fig. 1. A more-
detailed procedure for phase shifting is described inSubsection 2.B as well as in Ref. 24. Figures 4(a), 4(b),4(c), 4(d), and 4(e) are the phase-shifted interfero-grams at 0°, 90°, 180°, 270°, and 360°, respectively.We also obtained similar phase-shifted interfero-grams by changing the wavelength of the laser aswell as at a synthetic wavelength.
Fig. 3. Shearing interferograms at defocus positions correspond-ing to (a) �1, (b) �2, and (c) �. (d), (e), (f) Corresponding Fourierspectra.
The system was tested for 3-D shape measure-ment. A steplike object was fixed on the referenceplane and imaged onto the CCD camera. The fringepatterns were projected onto the discontinuous ob-ject. Figure 5(a) shows an example of the projectedfringe pattern, and the Fig. 5(b) shows the recon-structed 3-D shape of the object. For reconstructionwe used the simple Fourier-transform fringe analysistechnique.
4. Conclusions
A simple multifrequency spatial-carrier and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry has beendeveloped. We obtained changes in the spatial-carrierfrequency either by changing the wavelength of thelaser light or by slight defocusing. A synthetic inter-ferogram with low spatial-carrier frequency was ob-tained by use of the laser light of two wavelengthssimultaneously in a lateral shearing interferometer.We obtained the phase-shifted fringe patterns from thesame setup by simply moving the wedge plate in anin-plane parallel direction, using a linear translator.Our fringe projection system was tested for measure-ment of the three-dimensional shape of a discontinu-ous object. The present system has many advantages;e.g., it is a common-path interferometry and hence isinsensitive to external vibrations, it is compact, and itis relatively inexpensive.
The authors are thankful to M. Shoeb Faridi forhis help and D. P. Kothari, Director, Indian Insti-tute of Technology, Delhi, for his continual encour-agement.
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