- Email: [email protected]
Speckle shearography using a multiband light source
ARTICLE IN PRESS
Optics and Lasers in Engineering 40 (2003) 543–552
Speckle shearography using a multiband light source C. Falldorf*, W. Osten, E. Kolenovic BIAS, Bremer Institut fur . Angewandte Strahltechnik, Abt. Optical Metrology, Klagenfurter Str.2, D-28359 Bremen, Germany Received 22 January 2002; received in revised form 20 February 2002; accepted 21 February 2002
Abstract Applications for optical metrology usually use lasers as light sources, because of the excellent spatial and temporal coherence of the emitted light. By comparison, the demands of speckle shearography concerning the coherence of the light source are low. This enables certain white-light sources to be an option. In this paper, the feasibility of low coherence shearography is shown. For this purpose an experimental setup is designed, composed of a mercury arc lamp, a spatial filter and a Michelson interferometer. With respect to speckle shearography, important characteristics of the light source are described and the mercury arc lamp is shown to be suitable. Finally, some experimental investigations of an object under load are presented. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Optical metrology; White light; Speckle; Shearography; Interferometry
1. Introduction The objective of speckle shearography is to determine deformation gradients of objects under load [1,2]. This is done by comparing the phase differences of neighboring areas of the object surface for different object states. The spacing between the two correlated areas of the surface is called the shear. A common technique is to superpose coherently two images of the same surface with the corresponding images shifted by the shear. This can be realized, for example, by a simple wedge or a Michelson interferometer. The big potential of shearography for *Corresponding author. Tel.: +49-421-218-5063; fax: +49-421-218-5013. E-mail address: [email protected] (C. Falldorf). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 8 0 - 5
ARTICLE IN PRESS 544
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the nondestructive inspection of technical components with respect to material faults and structural imperfections was already proven on numerous examples. Especially during the past few years complete shearography systems have been implemented and applied successfully for the inspection of aircraft components [3,4]. Since speckle shearography is an interferometric technique, the use of spatially and temporally coherent light is essential. For that reason, lasers are commonly used as light sources for shearographic applications. However, compared to other coherent speckle techniques like ESPI or holographic interferometry, speckle shearography has lower demands concerning the coherence of the used light. This is due to the fact that only local areas of the object surface are compared. This means only short differences between the lengths of the optical paths have to be considered and consequently no excessive high temporal coherence is needed. On the other hand, the spatial coherence has to be in the magnitude of the shear. From this consideration it follows that certain white-light sources with a limited bandwidth should be an option for speckle shearography. Such a broadband light source should provide at least limited temporal coherence, spatial coherence and sufficient intensity for the obtained speckle field to be recorded by a CCD or a CMOS camera. One possible white light source, which provides the named properties, is the mercury arc lamp. It has three bright lines with a limited bandwidth in the visible spectrum. The measured full-width half-maximum (FWHM) for each line does not exceed 12 nm. Hence the coherence length should be in the order of several microns. The width of the arc can be chosen beneath 600 mm, thus resulting in a quasi point source with sufficient spatial coherence to obtain speckles. One of the lines resides near the ultraviolet spectra at 435.8 nm. The other two lines at 546.5 and 578.8 nm are well detected by the CCD. Therefore, in this paper an experimental setup is proposed which uses a mercury arc lamp to perform speckle shearography. The following section deals with the characterization of the light source. With respect to speckle shearography, the important features of the used lamp are examined. Those features are the spectrum as well as the temporal and the spatial coherence. Section 3 describes the experimental setup. Because multiple wavelengths are used, dispersion effects become an issue and must be considered. This is a key feature of the setup and will be explained in more detail. As a result, the underlying fringe modulation of the superposed wavelengths is shown. Finally, experimental investigations are presented in the last section. A clamped stripe of alloy had been observed while being stressed from the backside. The result of the measurement is a map of phase differences correlating with the object’s surface under stress.
2. Characterization of the light source The used mercury arc lamp is the OSRAM HBO 103 W/2. This lamp is especially designed for spectroscopic purposes. The spacing between the electrodes is 600 mm in the cold state and smaller while operating. A remarkable feature of that lamp is the very stable arc. This property and the low spacing of the electrodes makes it a suitable point-like light source.
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545
The measured visible spectrum of the lamp is shown in Fig. 1. The high pressure conditions in the lamp broadens the lines and let them appear Gaussian shaped, according to the Maxwellian velocity distribution. From the spectrum one can specify the FWHM of each line. Using the FWHM, it is possible to estimate the temporal coherence t of each line with the relation [5] 1 c c ¼ Dn ¼ ; ð1Þ t l Dl=2 l þ Dl=2 where l and Dl denote the wavelength and the spectral width specified from the FWHM of the line. This relation is only true for rectangular profiles. For Gaussian profiles a factor of ð2 ln 2=pÞ0:5 must be applied [5]. Consequently, the estimated coherence lengths Dsc;Gaussian are given by rffiffiffiffiffiffiffiffiffiffiffi 2 ln 2 : ð2Þ Dsc;Gaussian ¼ ct p The results are shown in Table 1. However, since the profiles of the lines are not exactly known, this calculation is just an estimation. To prove these results, the coherence lengths of the lines were measured by means of a Michelson
Radiant Intensity (W/sr/nm) per 1000 cd
0.12
0.1
0.08
0.06
0.04
0.02
0 420
470
520
570
620
Wavelength (nm)
Fig. 1. Spectral radiant intensity of the used mercury arc lamp HB0 103 W/2.
Table 1 Coherence lengths of the individual lines in the visible spectrum Wavelength l (nm) 435.8 546.5 578.8
Spectral width Dn (s1) *
6.29 1012 5.02* 1012 10.74* 1012
Dsc;Gaussian (mm)
Dsc;measured (mm)
32 40 18
3071 4271 1971
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546
interferometer. The relative path lengths of the interferometer arms Ds is varied and the visibility V of a certain fringe is calculated with V¼
Imax Imin ; Imax þ Imin
ð3Þ
where Imax and Imin denote the maximum and the minimum intensity of a certain fringe, respectively. The results for the lines at 535.8 and 546.5 nm are shown exemplarily in Figs. 2a and b. Given by the threshold of 1/e, the measured coherence lengths Dsc;measured are presented in Table 1. It can be seen that the measured
Visibility of the Fringes at 435.8 nm 1 0.9
Visibility (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
(a)
20
30
40
Position (µm)
Visibility of the Fringes at 546.5 nm 1 0.9
Visibility (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
(b)
10
20
30
40
50
Position (µm)
Fig. 2. The visibility of the fringes against the optical retardation.
ARTICLE IN PRESS C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
547
coherence lengths show good consistence with the estimated values. This verifies the lines to be Gaussian shaped. These results also show, that the maximum optical retardation with respect to interferometric measurement should not exceed 18 mm when using the line at 578.8 nm. An important consequence from this is a limited gradient of the objects surface which depends on the shear and the direction of illumination. Therefore, the surface of the object should be perpendicular to the bisecting line of the angle between the incident light and the direction of observation. Another important prerequisite for speckle shearography is the speckle field. To obtain speckles, a minimum spatial coherence is needed at least over an area as large as an Airy disc on the object surface. However, the spatial coherence does not only depend on the light source but on certain features of the experimental setup. For example, the position of a pinhole and the type of illumination must be considered. The used experimental setup will be presented in Section 3. To prove the spatial coherence of the light source within this setup, a subjective speckle field was produced by illuminating a rough surface. With the help of an aperture, the speckle size was chosen to fit the resolution of the camera target. To investigate the speckle statistics, quasi-monochromatic light was used [6]. Therefore, the light was filtered at 578.8 nm. The result is presented in Fig. 3a. A common method to verify a speckle field is to calculate the probability density function of the measured intensity distribution [7,8]. This is shown in Fig. 3b. The underlying black graph shows the probability density function PðIÞ as it is expected from theory 1 I PðIÞ ¼ exp ; ð4Þ /IS /IS where I and /IS denote the intensity and the mean intensity, respectively. The result shown in Fig. 3b is characteristic for a recorded speckle field. The measured probability density is in agreement with the theory, which proves the light to be spatially coherent at least over an area as large as an Airy disc on the object surface. The difference between the curves is due to the fact that the physical speckle field has been convoluted by the sensitivity profile of the camera targets pixels. This kind of low pass filtering decreases the probability for black and, therefore, changes the statistics for small intensities in the way shown by the measured density function in Fig. 3b [9]. With respect to shearographic applications, another interesting conclusion from this is that the width of the shear can be chosen at least within the diameter of an Airy disc.
3. Experimental setup The used experimental setup is shown in Fig. 4. The light of the mercury arc lamp is focused on a pinhole with a diameter of 200 mm. The chosen diameter is a trade-off between the needed spatial coherence and the intensity. After passing the pinhole, the beam is collimated and the rough surface of the object is illuminated to get a
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C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
Fig. 3. (a) Detected speckle field, (b) probability density function.
speckle field. The object itself is made up of a thin stripe of alloy with the surface being sandblasted. A screw is used to stress the clamped object from the backside. Finally, the scattered light is imaged to the camera target. The aperture is used to adjust the size of the speckles. A key feature of the setup is the Michelson interferometer which is used to create the shear. Because multiple wavelengths are used, dispersion effects become an issue whenever the optical retardation is varied. This happens at three different places of the setup: the object deformation, the tilted mirror of the Michelson interferometer and the phase shift which is used to apply phase sampling techniques.
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Fig. 4. (a, b). Experimental setup.
However, because of the different angular frequencies, the lines superpose incoherently [10].Thus the resulting intensity can be expressed with IðDsÞ ¼
3 X
2I0;i þ 2I0;i cos ðki DsÞ;
ð5Þ
i¼1
where Ds and ki denote the optical retardation and the wavenumber, respectively. As a first approach, we propose to lock the Michelson interferometer to zero order by adjusting the arms within the scope of nanometers. Next, we assume the line at 435.8 nm to be almost invisible to the CCD-target. Finally, we assume the dominating two lines at 546.5 and 578.8 nm to have a similar intensity of I0 : With
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these assumptions the superposition of the lines can be written as IðDsÞ ¼ 4I0 þ 2I0 cos ðk1 DsÞ þ 2I0 cosðk2 DsÞ k1 þ k2 k1 k2 ¼ 4I0 þ 4I0 cos Ds cos Ds : 2 2
ð6Þ
Since the two considered lines are adjacent in the spectrum, k1 is in the order of k2 : In this case, it is seen from Eq. (6), that the resulting fringe modulation is a superposition of a fast and a slow varying cosine, respectively. Hence several fringes should appear until the fringe modulation vanishes due to dispersion effects. The wavenumber of the resulting fringes is obtained by averaging k1 and k2 : k1 þ k2 : ð7Þ kave ¼ 2 Fig. 5 shows the measured intensity with respect to the optical retardation, which was varied by moving one mirror of the Michelson interferometer. The underlying dashed graph shows the intensity as it is expected following Eq. (6). From Fig. 5 it can be seen that there is a moderate fringe modulation of at least 1/e for approximately 4 fringes. Considering both directions and the center peak, 9 fringes are available for interferometric applications. The difference between the measured intensity and the theoretically obtained curve is the consequence of the neglected line at 435.8 nm. However, an important result with respect to phase sampling techniques is that the period of the measured data is in good agreement with the theory. Thus the phase shift can be calculated using the wavenumber kave derived from Eq. (7).
Intensity vs Optical Retardation 1 Measured Theory
0.9
Intensity (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 4π
8π
Phase difference Fig. 5. Fringes of the mercury arc lamp.
12 π
16 π
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4. Experimental investigations Fig. 6 shows experimental results which were obtained using the setup presented in the previous section. The object under test was a thin stripe of alloy with the surface being sandblasted. It was clamped and stressed from the backside with a screw. The thickness of the stripe was 100 mm and the open diameter of the clamp amounted 12 mm. A simple phase sampling technique was used to obtain the modulo 2p phase maps. Four frames were taken with the phase stepped by p=2 for each frame. The phase map j was calculated modulo 2p from jðx; yÞ mod 2p ¼ tan
1
I4 ðx; yÞ I2 ðx; yÞ ; I1 ðx; yÞ I3 ðx; yÞ
ð8Þ
where I14 denote the intensity distribution of the taken frames. Fig. 6a shows the object with a small deformation. In Fig. 6b the same scenario is seen with the object under higher load. Finally, Figs. 6c and d show the measured data filtered and unwrapped, respectively.
Fig. 6. (a) Low deformation, (b) higher deformation, (c) filtered phase map, (d) unwrapped phase map, (a–d) experimental results: two object deformations, filtered and unwrapped.
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5. Conclusion The previous sections proved the mercury arc lamp to be a suitable light source for shearographic applications. It was characterized with respect to interferometric applications and used in a shearographic setup. Finally, experimental results were presented. Current limitations are the small deformation range and the low intensity compared to the laser. Thus only small objects can be investigated. The small deformation range is a direct consequence of the fringe modulation, which is limited by dispersion effects as shown in Section 3. This could be improved by a dispersion compensation. With respect to the deformation, another limitation is the small coherence length of the line at 578.8 nm. Compared to the laser, the intensity of the mercury arc lamp is low. This is an inherent vice of the lamp and cannot be improved. Therefore, it is necessary to use more sensitive sensors when larger objects are investigated.
References [1] Hung YY. Image-shearing camera for direct measurement of strains. Appl Opt 1979;18:1046–51. [2] Kreis Th. holographic interferometry. Berlin: Akademie Verlag, 1996. p. 263. [3] Ettemeyer A. Fast, reliable development of materials and components using modern laser speckle techniques. Laser Opt Oct. 2000; 62–6. [4] Osten W, Kalms M, Juptner . W, Tober G, Bisle D, Scherling D. A shearography system for the testing of large scale aircraft components taking into account non-cooperative surfaces. Proc SPIE 2000;4101B:432–8. [5] Saleh BEA, Teich MC. Fundamentals of photonics. New York: Wiley, Kap. 10. [6] D.andliker R. Heterodyne holographic interferometry. In: E. Wolf, editor. Progress in optics, vol. XVII. Amsterdam: North-Holland, 1980. [7] Goodman JW. Statistical properties of laser speckle patterns. In: Dainty JC, editor. Laser speckle and related phenomena. Berlin: Springer; 1975. p. 9–75. [8] Kadono H, Asakura T. Statistical properties of the speckle phase in the optical imaging system, JOSA, 1985. [9] Osten W. Digital processing and evaluation of fringe patterns in optical metrology and nondestructive testing. In: Laermann KH, editor. Optical methods in experimental solid mechanics. New York: Springer Wien; 2000. p. 289. [10] Born M, Wolf E. Principles of optics, Cambridge: Cambridge University Press. 1997. p. 320.
Optics and Lasers in Engineering 40 (2003) 543–552
Speckle shearography using a multiband light source C. Falldorf*, W. Osten, E. Kolenovic BIAS, Bremer Institut fur . Angewandte Strahltechnik, Abt. Optical Metrology, Klagenfurter Str.2, D-28359 Bremen, Germany Received 22 January 2002; received in revised form 20 February 2002; accepted 21 February 2002
Abstract Applications for optical metrology usually use lasers as light sources, because of the excellent spatial and temporal coherence of the emitted light. By comparison, the demands of speckle shearography concerning the coherence of the light source are low. This enables certain white-light sources to be an option. In this paper, the feasibility of low coherence shearography is shown. For this purpose an experimental setup is designed, composed of a mercury arc lamp, a spatial filter and a Michelson interferometer. With respect to speckle shearography, important characteristics of the light source are described and the mercury arc lamp is shown to be suitable. Finally, some experimental investigations of an object under load are presented. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Optical metrology; White light; Speckle; Shearography; Interferometry
1. Introduction The objective of speckle shearography is to determine deformation gradients of objects under load [1,2]. This is done by comparing the phase differences of neighboring areas of the object surface for different object states. The spacing between the two correlated areas of the surface is called the shear. A common technique is to superpose coherently two images of the same surface with the corresponding images shifted by the shear. This can be realized, for example, by a simple wedge or a Michelson interferometer. The big potential of shearography for *Corresponding author. Tel.: +49-421-218-5063; fax: +49-421-218-5013. E-mail address: [email protected] (C. Falldorf). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 8 0 - 5
ARTICLE IN PRESS 544
C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
the nondestructive inspection of technical components with respect to material faults and structural imperfections was already proven on numerous examples. Especially during the past few years complete shearography systems have been implemented and applied successfully for the inspection of aircraft components [3,4]. Since speckle shearography is an interferometric technique, the use of spatially and temporally coherent light is essential. For that reason, lasers are commonly used as light sources for shearographic applications. However, compared to other coherent speckle techniques like ESPI or holographic interferometry, speckle shearography has lower demands concerning the coherence of the used light. This is due to the fact that only local areas of the object surface are compared. This means only short differences between the lengths of the optical paths have to be considered and consequently no excessive high temporal coherence is needed. On the other hand, the spatial coherence has to be in the magnitude of the shear. From this consideration it follows that certain white-light sources with a limited bandwidth should be an option for speckle shearography. Such a broadband light source should provide at least limited temporal coherence, spatial coherence and sufficient intensity for the obtained speckle field to be recorded by a CCD or a CMOS camera. One possible white light source, which provides the named properties, is the mercury arc lamp. It has three bright lines with a limited bandwidth in the visible spectrum. The measured full-width half-maximum (FWHM) for each line does not exceed 12 nm. Hence the coherence length should be in the order of several microns. The width of the arc can be chosen beneath 600 mm, thus resulting in a quasi point source with sufficient spatial coherence to obtain speckles. One of the lines resides near the ultraviolet spectra at 435.8 nm. The other two lines at 546.5 and 578.8 nm are well detected by the CCD. Therefore, in this paper an experimental setup is proposed which uses a mercury arc lamp to perform speckle shearography. The following section deals with the characterization of the light source. With respect to speckle shearography, the important features of the used lamp are examined. Those features are the spectrum as well as the temporal and the spatial coherence. Section 3 describes the experimental setup. Because multiple wavelengths are used, dispersion effects become an issue and must be considered. This is a key feature of the setup and will be explained in more detail. As a result, the underlying fringe modulation of the superposed wavelengths is shown. Finally, experimental investigations are presented in the last section. A clamped stripe of alloy had been observed while being stressed from the backside. The result of the measurement is a map of phase differences correlating with the object’s surface under stress.
2. Characterization of the light source The used mercury arc lamp is the OSRAM HBO 103 W/2. This lamp is especially designed for spectroscopic purposes. The spacing between the electrodes is 600 mm in the cold state and smaller while operating. A remarkable feature of that lamp is the very stable arc. This property and the low spacing of the electrodes makes it a suitable point-like light source.
ARTICLE IN PRESS C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
545
The measured visible spectrum of the lamp is shown in Fig. 1. The high pressure conditions in the lamp broadens the lines and let them appear Gaussian shaped, according to the Maxwellian velocity distribution. From the spectrum one can specify the FWHM of each line. Using the FWHM, it is possible to estimate the temporal coherence t of each line with the relation [5] 1 c c ¼ Dn ¼ ; ð1Þ t l Dl=2 l þ Dl=2 where l and Dl denote the wavelength and the spectral width specified from the FWHM of the line. This relation is only true for rectangular profiles. For Gaussian profiles a factor of ð2 ln 2=pÞ0:5 must be applied [5]. Consequently, the estimated coherence lengths Dsc;Gaussian are given by rffiffiffiffiffiffiffiffiffiffiffi 2 ln 2 : ð2Þ Dsc;Gaussian ¼ ct p The results are shown in Table 1. However, since the profiles of the lines are not exactly known, this calculation is just an estimation. To prove these results, the coherence lengths of the lines were measured by means of a Michelson
Radiant Intensity (W/sr/nm) per 1000 cd
0.12
0.1
0.08
0.06
0.04
0.02
0 420
470
520
570
620
Wavelength (nm)
Fig. 1. Spectral radiant intensity of the used mercury arc lamp HB0 103 W/2.
Table 1 Coherence lengths of the individual lines in the visible spectrum Wavelength l (nm) 435.8 546.5 578.8
Spectral width Dn (s1) *
6.29 1012 5.02* 1012 10.74* 1012
Dsc;Gaussian (mm)
Dsc;measured (mm)
32 40 18
3071 4271 1971
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546
interferometer. The relative path lengths of the interferometer arms Ds is varied and the visibility V of a certain fringe is calculated with V¼
Imax Imin ; Imax þ Imin
ð3Þ
where Imax and Imin denote the maximum and the minimum intensity of a certain fringe, respectively. The results for the lines at 535.8 and 546.5 nm are shown exemplarily in Figs. 2a and b. Given by the threshold of 1/e, the measured coherence lengths Dsc;measured are presented in Table 1. It can be seen that the measured
Visibility of the Fringes at 435.8 nm 1 0.9
Visibility (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
(a)
20
30
40
Position (µm)
Visibility of the Fringes at 546.5 nm 1 0.9
Visibility (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
(b)
10
20
30
40
50
Position (µm)
Fig. 2. The visibility of the fringes against the optical retardation.
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547
coherence lengths show good consistence with the estimated values. This verifies the lines to be Gaussian shaped. These results also show, that the maximum optical retardation with respect to interferometric measurement should not exceed 18 mm when using the line at 578.8 nm. An important consequence from this is a limited gradient of the objects surface which depends on the shear and the direction of illumination. Therefore, the surface of the object should be perpendicular to the bisecting line of the angle between the incident light and the direction of observation. Another important prerequisite for speckle shearography is the speckle field. To obtain speckles, a minimum spatial coherence is needed at least over an area as large as an Airy disc on the object surface. However, the spatial coherence does not only depend on the light source but on certain features of the experimental setup. For example, the position of a pinhole and the type of illumination must be considered. The used experimental setup will be presented in Section 3. To prove the spatial coherence of the light source within this setup, a subjective speckle field was produced by illuminating a rough surface. With the help of an aperture, the speckle size was chosen to fit the resolution of the camera target. To investigate the speckle statistics, quasi-monochromatic light was used [6]. Therefore, the light was filtered at 578.8 nm. The result is presented in Fig. 3a. A common method to verify a speckle field is to calculate the probability density function of the measured intensity distribution [7,8]. This is shown in Fig. 3b. The underlying black graph shows the probability density function PðIÞ as it is expected from theory 1 I PðIÞ ¼ exp ; ð4Þ /IS /IS where I and /IS denote the intensity and the mean intensity, respectively. The result shown in Fig. 3b is characteristic for a recorded speckle field. The measured probability density is in agreement with the theory, which proves the light to be spatially coherent at least over an area as large as an Airy disc on the object surface. The difference between the curves is due to the fact that the physical speckle field has been convoluted by the sensitivity profile of the camera targets pixels. This kind of low pass filtering decreases the probability for black and, therefore, changes the statistics for small intensities in the way shown by the measured density function in Fig. 3b [9]. With respect to shearographic applications, another interesting conclusion from this is that the width of the shear can be chosen at least within the diameter of an Airy disc.
3. Experimental setup The used experimental setup is shown in Fig. 4. The light of the mercury arc lamp is focused on a pinhole with a diameter of 200 mm. The chosen diameter is a trade-off between the needed spatial coherence and the intensity. After passing the pinhole, the beam is collimated and the rough surface of the object is illuminated to get a
ARTICLE IN PRESS 548
C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
Fig. 3. (a) Detected speckle field, (b) probability density function.
speckle field. The object itself is made up of a thin stripe of alloy with the surface being sandblasted. A screw is used to stress the clamped object from the backside. Finally, the scattered light is imaged to the camera target. The aperture is used to adjust the size of the speckles. A key feature of the setup is the Michelson interferometer which is used to create the shear. Because multiple wavelengths are used, dispersion effects become an issue whenever the optical retardation is varied. This happens at three different places of the setup: the object deformation, the tilted mirror of the Michelson interferometer and the phase shift which is used to apply phase sampling techniques.
ARTICLE IN PRESS C. Falldorf et al. / Optics and Lasers in Engineering 40 (2003) 543–552
549
Fig. 4. (a, b). Experimental setup.
However, because of the different angular frequencies, the lines superpose incoherently [10].Thus the resulting intensity can be expressed with IðDsÞ ¼
3 X
2I0;i þ 2I0;i cos ðki DsÞ;
ð5Þ
i¼1
where Ds and ki denote the optical retardation and the wavenumber, respectively. As a first approach, we propose to lock the Michelson interferometer to zero order by adjusting the arms within the scope of nanometers. Next, we assume the line at 435.8 nm to be almost invisible to the CCD-target. Finally, we assume the dominating two lines at 546.5 and 578.8 nm to have a similar intensity of I0 : With
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550
these assumptions the superposition of the lines can be written as IðDsÞ ¼ 4I0 þ 2I0 cos ðk1 DsÞ þ 2I0 cosðk2 DsÞ k1 þ k2 k1 k2 ¼ 4I0 þ 4I0 cos Ds cos Ds : 2 2
ð6Þ
Since the two considered lines are adjacent in the spectrum, k1 is in the order of k2 : In this case, it is seen from Eq. (6), that the resulting fringe modulation is a superposition of a fast and a slow varying cosine, respectively. Hence several fringes should appear until the fringe modulation vanishes due to dispersion effects. The wavenumber of the resulting fringes is obtained by averaging k1 and k2 : k1 þ k2 : ð7Þ kave ¼ 2 Fig. 5 shows the measured intensity with respect to the optical retardation, which was varied by moving one mirror of the Michelson interferometer. The underlying dashed graph shows the intensity as it is expected following Eq. (6). From Fig. 5 it can be seen that there is a moderate fringe modulation of at least 1/e for approximately 4 fringes. Considering both directions and the center peak, 9 fringes are available for interferometric applications. The difference between the measured intensity and the theoretically obtained curve is the consequence of the neglected line at 435.8 nm. However, an important result with respect to phase sampling techniques is that the period of the measured data is in good agreement with the theory. Thus the phase shift can be calculated using the wavenumber kave derived from Eq. (7).
Intensity vs Optical Retardation 1 Measured Theory
0.9
Intensity (normalized)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 4π
8π
Phase difference Fig. 5. Fringes of the mercury arc lamp.
12 π
16 π
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4. Experimental investigations Fig. 6 shows experimental results which were obtained using the setup presented in the previous section. The object under test was a thin stripe of alloy with the surface being sandblasted. It was clamped and stressed from the backside with a screw. The thickness of the stripe was 100 mm and the open diameter of the clamp amounted 12 mm. A simple phase sampling technique was used to obtain the modulo 2p phase maps. Four frames were taken with the phase stepped by p=2 for each frame. The phase map j was calculated modulo 2p from jðx; yÞ mod 2p ¼ tan
1
I4 ðx; yÞ I2 ðx; yÞ ; I1 ðx; yÞ I3 ðx; yÞ
ð8Þ
where I14 denote the intensity distribution of the taken frames. Fig. 6a shows the object with a small deformation. In Fig. 6b the same scenario is seen with the object under higher load. Finally, Figs. 6c and d show the measured data filtered and unwrapped, respectively.
Fig. 6. (a) Low deformation, (b) higher deformation, (c) filtered phase map, (d) unwrapped phase map, (a–d) experimental results: two object deformations, filtered and unwrapped.
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5. Conclusion The previous sections proved the mercury arc lamp to be a suitable light source for shearographic applications. It was characterized with respect to interferometric applications and used in a shearographic setup. Finally, experimental results were presented. Current limitations are the small deformation range and the low intensity compared to the laser. Thus only small objects can be investigated. The small deformation range is a direct consequence of the fringe modulation, which is limited by dispersion effects as shown in Section 3. This could be improved by a dispersion compensation. With respect to the deformation, another limitation is the small coherence length of the line at 578.8 nm. Compared to the laser, the intensity of the mercury arc lamp is low. This is an inherent vice of the lamp and cannot be improved. Therefore, it is necessary to use more sensitive sensors when larger objects are investigated.
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