- Email: [email protected]
Surface roughness measurement by speckle contrast under the illumination of light with arbitrary spectral profile
ARTICLE IN PRESS Optics and Lasers in Engineering 48 (2010) 774–778
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Surface roughness measurement by speckle contrast under the illumination of light with arbitrary spectral profile Lioudmila Tchvialeva a, Igor Markhvida a, Haishan Zeng a,b, David I. McLean a, Harvey Lui a,b, Tim K. Lee a,b,c, a
Photomedicine Institute, Department of Dermatology and Skin Science, University of British Columbia and Vancouver Coastal Health Research Institute, Vancouver, Canada V5Z 4E8 b Departments of Cancer Control Research and Cancer Imaging, BC Cancer Agency, Vancouver, Canada V5Z 1L3 c School of Computing Science, Simon Fraser University, Burnaby, Canada V5A 1S6
a r t i c l e in f o
a b s t r a c t
Article history: Received 28 November 2009 Accepted 12 March 2010 Available online 2 April 2010
Quantification of surface roughness greater than a micron is desirable for many industrial and biomedical applications. Polychromatic speckle contrast has been shown theoretically to be able to detect such roughness range using an appropriate light source with a Gaussian spectral shape. In this paper, we extend the theory to arbitrary spectral profile by formulating speckle contrast as a function of spectral profile, surface roughness, and the geometry of speckle formation. Under a far-field set-up, the formulation can be simplified and a calibration curve for contrast and roughness can be calculated. We demonstrated the technique using a blue diode laser with a set of 20 metal surface roughness standards in the range 1–73 mm, and found that the method worked well with both Gaussian and non-Gaussian surfaces. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Surface roughness Polychromatic speckle contrast Far-field Fraunhofer zone
1. Introduction Optical techniques for measuring surface roughness are advantageous for their noncontact and nondestructive properties. 3D mapping techniques that utilize scanning or image processing usually require complex hardware. On the other hand, techniques that measure only roughness statistics can be based on light scattering using low-cost, simple devices with rapid data acquisition times. The analysis of speckles that arise from coherent light scattered by a surface belongs to the latter. Although the theoretical foundations of speckle methods have been established few decades ago [1], this methodology has not yet been implemented into practical instrumentation due to technological constraints. Recent advances in light sources and registration devices have rekindled interest in speckle-based techniques. Using speckle for surface roughness can be classified in the following ways. One class of speckle techniques utilizes two or more speckle fields and requires a complex hardware design [2–4]. In general these techniques are not suitable for industrial applications or biological tissue examination. Another class overcomes this
Corresponding author at: BC Cancer Research Centre, Cancer Control Research Program, 675 West 10th Avenue, Vancouver, BC, Canada V5Z 1L3. Tel.: + 1 604 675 8053; fax: + 1 604 675 8180. E-mail address: [email protected] (T.K. Lee).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.03.004
disadvantage by analyzing simple speckle patterns of scattered light captured with a camera. This method can be further subclassified: analyzing either the (1) spatial [5–8] or (2) intensity [9–17] statistics for the speckle patterns. The main drawbacks of the spatial statistics approach are the difficulty of calibration and the lack of theoretical formulations, which result in qualitative ‘‘ad-hoc’’ roughness assessments. On the other hand, intensity statistics such as speckle contrast [9] is based on well-studied theories. Monochromatic speckle contrast has been successfully utilized for submicron range roughness measurements [10–12] for weak-scattering surface conditions where wavelength determines the physical upper limit for measurements. Monochromatic speckle has also been applied over a larger roughness range in two studies. However, such attempts yielded only qualitative results with a complex two-scale surface structure [13] in addition to some unexplained data that conflicted with the theory [14]. Polychromatic speckle contrast has been shown theoretically [15,16] for measuring roughness in ranges larger than a micron as long as an appropriate light source is used. Coherence length is the scale of measurement, and this fact was qualitatively demonstrated in [17]. The theoretical formulations in [15,16] rely on a Gaussian spectral shape, and the formulations to our best knowledge have not been realized in practice. In this paper, we describe how to measure roughness by polychromatic speckle contrast under illumination with an arbitrary spectral profile. In the theoretical part we present a formula for the numerical calculation of a calibration curve relating contrast with roughness. We also demonstrate a practical realization of the
ARTICLE IN PRESS L. Tchvialeva et al. / Optics and Lasers in Engineering 48 (2010) 774–778
775
technique using diode lasers that are inexpensive, reliable, and widely available. As an example we calculate the calibration curves separately for a red and a blue laser diode, each with its own multipeak, complex spectrum. We validate our technique using metal roughness standards in the range 1–80 mm. Though the set-up we used is very simple, we emphasize the design principles of the device for achieving consistent results.
2. Theory
Fig. 1. Influence of width of function on integral. The three Gaussian functions can represent any of the three bell-shaped functions.
Contrast C of any speckle pattern is defined as [9] C¼
sI /IS
,
ð1Þ
where /yS denotes an ensemble averaging, and sI is the standard deviation of light intensity I, with sI2 being the variance:
s2I ¼ /I2 S/IS2 :
ð2Þ
By illuminating a surface with a polychromatic light source that has a finite spectrum as well as a finite temporal coherence length, the intensity of polychromatic speckles is the sum of the monochromatic speckle pattern intensities: Z 1 FðkÞIðx,kÞdk, ð3Þ IðxÞ ¼ 0
where k is wave number, F(k) is the spectral line profile of the illuminating light, and x is a vector in the observation plane. Eq. (3) implies that the registration time is much greater than the time of coherence. In other words, speckle patterns created by individual wavelengths are incoherent and we summarize the intensity. The behavior of I(x) depends on many factors. In some areas of the observation plane, intensity I(x, k) for all k will have the same distribution and the contrast of the resultant speckle pattern will be same as the contrast of the single monochromatic pattern. In other areas, the patterns will be shifted and their sum produces a smoothed speckle pattern with a reduced contrast. The second moment of the intensity I(x) can be calculated using Eq. (3) to yield Z 1Z 1 Fðk1 ÞFðk2 Þ/Iðx,k1 ÞIðx,k2 ÞSdk1 dk2 : ð4Þ /I2 ðxÞS ¼ 0
0
Calculating the variance according to Eqs. (2)–(4) we obtain Z 1Z 1 s2I ¼ /I2 S/IS2 ¼ Fðk1 ÞFðk2 Þ½/Iðx,k1 ÞIðx,k2 ÞS 0
0
/Iðx,k1 ÞS/Iðx,k2 ÞS dk1 dk2
ð5Þ
It has been shown in [18] that Eqs. (3) and (5) can be transformed to Z 1 Z 1 Z s2I ¼ Fðk1 ÞFðk1 þ DkÞdk1 jMDh ðDkÞ2 j exp½iDkRdSj2 dDk: 0
1
ð6Þ /IðxÞS ¼ S
Z
1
FðkÞdk,
ð7Þ
of intensity depending on the light source spectrum F(k), the second term R(Dk) relates to surface roughness, and the third term D(Dk)¼|R exp[ iDkR]dS|2/S2 represents the geometry of speckle formation. Eq. (8) has an important methodological meaning: it shows that contrast depends on three factors, namely, light coherence, surface roughness, and geometry. A group of techniques based on contrast measurement can be developed to determine any of these factors. As a rule, functions G(Dk), R(Dk), and D(Dk) have a bell-shape form (or limited width). To simplify the development of a measurement technique, relationships among the scale of coherence length, roughness, and optical paths can be chosen in such a way to allow the exclusion of unnecessary dependencies in Eq. (8) and/or varying the sensitivity of the contrast measurement technique to the appropriate parameter. This concept is illustrated in Fig. 1 where three bell-shaped functions are shown and an integral (contrast) depends mainly on width of function 3. Eq. (8) can be simplified for the purpose of roughness measurement. We will use only the central part of the speckle pattern x E0 to obtain the relationship between contrast and surface roughness. If we configure an optical set-up in a way that the illuminated surface area is smaller than coherence zone [9,18], the third integral in Eq. (8) approaches S2. For example, this condition will be satisfied when a parallel beam illuminates a spot q, which is in a distance z away from the camera, and Dkq2/z51. This is true for free geometry speckle in the area of Fraunhofer diffraction (zFraunhofer) and even for a distance z which is n times smaller than zFraunhofer, where n ¼lc/l, lc is a coherence length, and l is the average wavelength for the spectrum. If the surface height deviation has a normal distribution with a root mean square (rms) height deviation sh, the characteristics function of the surface height becomes [9] 2
MDh ðDkÞ ¼ eðð2sh Þ
Dk2 =2Þ
:
ð9Þ
Finally, combining Eq. (1) and Eqs. (6)–(8), the polychromatic speckle contrast can be expressed as R1 R1 2 0 ð 0 FðkÞFðk þ DkÞdkÞexpðð2sh DkÞ2 ÞdDk : ð10Þ C 2 ðsh Þ ¼ R1 ð 0 FðkÞdkÞ2 Knowing the emission light source spectrum F(k) and performing a simple numerical calculation of Eq. (10), the calibration curve for the contrast C vs. the RMS roughness sh can be obtained.
0
where Dk¼k2 k1, R is the full geometrical path of elementary waves from the source to the observation plane (excluding the additional phase shift by the diffuser), S is the illuminated spot area, and MDh ðDkÞ is the characteristic function of the surface height. Substituting Eqs. (6) and (7) into Eq. (1) results in Z 1 GðDkÞRðDkÞDðDkÞdDk: ð8Þ C2 ¼ 1
R 1 In Eq. (8) the first term GðDkÞ ¼ 0 Fðk1 ÞFðk1 þ DkÞdk1 = 2 R 1 inside the integral is the autocorrelation function 0 FðkÞdk
3. Calibration curves for blue and red diode lasers One of the most commonly used low-coherent light sources is the diode laser. A typical diode laser resonator is a Fabry–Perot interferometer with multi-peak emission spectrum [19]. In Fig. 2, spectra of two diode lasers are shown: (a) a blue diode laser (20 mW, BWB-405-20E, B&W Tek, Inc., USA) and (b) a red fiber-coupled diode laser (5 mW, 57PNL054/P4/SP, Melles Griot Inc., USA). These spectra were obtained by a Bomem DA8 Fourier transform interferometer system, with a quartz (visible) beamsplitter and a Si diode detector.
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To find a calibration curve we performed a numerical integration of Eq. (10) with the experimental diode laser spectra converted to F(k). The calculated dependence of contrast vs roughness is shown in Fig. 3(a). It should be noted that these curves do not describe contrast behavior in the submicron range. For comparison, we include the calibration curve for a light source with a Gaussian profile and with the same halfwidth as the red laser spectrum envelope in Fig. 3a (dotted line). Despite the similar trend for calibration curves generated by the Gaussian [15,16] and non-Gaussian sources, it is evident that the former curve cannot be used for precise measurements. For example, a speckle contrast of 0.8 corresponds to roughness about 60 mm using the curve calculated from (10) while the Gaussian spectrum example results in a roughness value of about 150 mm. In Fig. 3(b), we show the first derivates of the normalized calibration curves vs. roughness to find the highest measurement sensitivity and to identify the effective measurable roughness range for our light sources. The blue diode works better for roughness of around 20 mm, while for the red diode it is about 40 mm. The accuracy of the contrast measurements that we obtained in experiments was approximately 0.01. Using the first derivative we can estimate that the best accuracy we can expect is about 1 mm for the blue laser and about 2 mm for the red. The blue diode provides better accuracy overall, and both lasers can be effectively used for roughness ranging up to 100 mm.
For the experimental targets, we chose a set of 20 metal surface roughness standards with three types of finishing: gridblasted (G-6, Gar Electroforming Division, Electroforming, Inc., USA), spark erosion (Microshurf #331, Rubert+ Company Ltd., UK), and casting (Microshurf #334, Rubert +Company Ltd., UK). We scanned these surfaces with a WYKO profilometer and calculated their rms roughness. The set of 20 surfaces provided a roughness range 1–73 mm. Analyzing the histograms of roughness showed that most of the surfaces had a Gaussian distribution. Some exceptions were noted for the large scales; Fig. 4 illustrates the most significant difference. While Microshuft #331 had a Gaussian distribution, the Microshuft #334 histogram deviated from a Gaussian function significantly. Therefore, we were able to test our method with both Gaussian and nonGaussian surfaces. The laser speckle set-up is shown in Fig. 5. A parallel laser beam illuminated a surface of interest. The scattered light was acquired by a monochrome CCD camera (mvBlueFOX-M124G, Matrix Vision GmbH, Germany) without an imaging lens. The Pellicle beam splitter provided a 01 beam incidence and speckle pattern observation in the specular direction. Although the optical configuration is very simple, the principles described next must be observed in order to achieve proper measurements. The key design points are proper level of intensity, well-placed distances amongst the source, target and the camera, appropriate speckle size, and reduction of camera dark current and ambient illumination. We discovered that the average intensity can be estimated as /IS¼2m/ln N, where m is the depth of a camera pixel and N is the number of speckles. The derivation of this equation is out of the scope of this paper and we are preparing another article to fully analyze the effect of the camera resolution on the average intensity. Essentially, the typical value of /IS for a 8-bit camera
4. Experiment
Intensity (arb. units)
Intensity (arb. units)
In this section, we discuss the set-up of the experiment and compare the experimental results with the calculated calibration curves.
407.1 407.3 407.5 407.7 407.9 408.1
657.5
658
Wavelength, nm
658.5
659
659.5
Wavelength, nm
1
0.008
0.8
0.006
Slope, 1/µm
Speckle Contrast
Fig. 2. Actual diode lasers spectral profiles: (a) blue diode laser and (b) red diode laser. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0.6 0.4 0.2
0.004 0.002 0.000
0 0
50 100 150 Roughness, µm
200
0
50
100 150 Roughness, µm
200
Fig. 3. (a) Calibration curves and (b) their first derivatives, which show the sensitivity of the corresponding calibration curve. Solid and dash lines represent the calibration curves for the red and blue lasers, respectively. Dotted line on (a) corresponds to the calibration curve generated by a Gaussian source. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ARTICLE IN PRESS L. Tchvialeva et al. / Optics and Lasers in Engineering 48 (2010) 774–778
777
1.1
8
1.0
MS 331 MS 334 Gaussian
6
0.9 0.8 contrast
pdf
0.7
4
0.6
Red theory Blue theory 331 Red 334 Red G6 Red 331 Blue 334 Blue G6 Blue
0.5 0.4
2
0.3 0.2
0 -0.25
-0.15
-0.05 0.05 Height, µm
0.15
0.25
0.1 0.0
Fig. 4. Comparing the probability density functions of the roughness for Microshuft #331(sh ¼ 62.5 mm) and Microshuft #334(sh ¼ 62.5 mm) with a Gaussian function(sh ¼ 62.5 mm).
0
20
40
60
roughness, µm Fig. 6. Comparison of calibration curves with experimental validation.
CCD Camera Polarizer
Beam splitter laser
Collimator
ensemble averaging can be replaced by spatial averaging which in turn means that we can use data from each pixel to calculate the average intensity and its standard deviation. The experimental procedure described in previous paragraphs was applied to a set of 20 metal standards. The measured contrasts are plotted against the rms roughness of the standards in Fig. 6. A good agreement between theory and experiment is observed for each surface finishing type. As expected, the smooth surfaces with roughness less than 8 mm form speckle patterns with contrasts close to unity.
5. Conclusion Rough surface Fig. 5. The speckle geometry set-up.
should be around 30. For metal standards, it is a good idea to employ a polarizer to prevent overexposure. The camera–surface distance z and the beam width q determine the size of speckles s which should be in the order of lz/q. Reducing the speckle size leads to the reduction of the contrast because of spatial averaging effect over the camera pixels. Similarly, a small amount of over-sized speckle also causes lower accuracy in contrast measurement. The minimum speckle should be at least twice the pixel size; speckle size is typically adjusted to be about 5–10 CCD camera pixels which can be easily controlled by analyzing the image. The surface to camera distance z should be large enough for the far-field (Fraunhofer) zone. As mentioned earlier it can be decreased n ¼lc/l times if necessary. In our experiment the optimal distance z was 270 mm; the illuminated spot diameters were 3 mm for the blue laser and 4.5 mm for the red laser. These settings provided the average speckle size of 60 and 80 mm for the blue and red, respectively. Camera dark current and ambient illumination can be removed if we subtracted an image without laser exposure from the speckle images. It should be noted that if the above conditions are achieved, the speckle pattern will be spatially uniform and
Using inexpensive diode lasers allows one to determine the surface roughness in the tens of micron range (10–100 mm) by simply measuring polychromatic speckle contrast. It was shown that models based on Gaussian spectra can be used only for qualitative estimations. For precise measurements the calibration curve for contrast should be calculated by Eq. (10). Despite the assumption of Gaussian height distribution used in the model, our experiments showed that the technique can be applied for a wide class of surfaces characterized by both Gaussian and non-Gaussian height distributions. We believe that this study contributes to the industrial application of speckle contrast, and will be of interest for surface metrology applications.
Acknowledgements This work was supported in part by Grants from the Canadian Institutes of Health Research, the Natural Sciences and Engineering Research Council of Canada, Canadian Dermatology Foundation, UBC & VGH Hospital Foundation, and UBC Faculty of Medicine. Also we would like to thank Dr. M.L. Thewalt (Simon Fraser University, Burnaby, Canada) for measuring the emission spectra of the red and the blue diode lasers and Mr. S. Yick (Institute for Fuel Cell Innovation, Vancouver, Canada) for measuring the target surface roughness.
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Disclosure statement The funding agencies have no role in study design; in the collection, analysis, and interpretation of data; in the writing of the report; and in the decision to submit the paper for publication. References [1] Briers J. Surface roughness evaluation. In: Sirohi RS, editor. Speckle metrology. CRC Press; 1993. [2] Peters J, Schoene A. Nondestructive evaluation of surface roughness by speckle correlation techniques. SPIE 1998;3399:45–56. [3] Death DL, Eberhardt JE, Rogers CA. Transparency effects on powder speckle decorrelation. Optics Express 2000;6:202–12. [4] Fricke-Begemann T, Hinsch K. Measurement of random processes at rough surfaces with digital speckle correlation. Journal of Optical Society of America A: Optics, Image Science, and Vision 2004;21:252–62. [5] Lehmann P. Aspect ratio of elongated polychromatic far-field speckles of continuous and discrete spectral distribution with respect to surface roughness characterization. Applied Optics 2002;41:2008–14. [6] Lu R-S, Tian G-Y, Gledhill D, Ward S. Grinding surface roughness measurement based on the co-occurrence matrix of speckle pattern texture. Applied Optics 2006;45:8839–47. [7] Li Z, Li H, Qiu Y. Fractal analysis of laser speckle for measuring roughness. SPIE 2006;6027:60271S. [8] Lehmann P. Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination. Applied Optics 1999;38:1144–52.
[9] Goodman JW. Speckle phenomena in optics: theory and application. Roberts and Company Publishers; 2006. [10] Fujii H, Asakura T. Roughness measurements of metal surfaces using laser speckle. JOSA 1977;67:1171–6. [11] Cheng C, Liu C, Zhang N, Jia T, Li R, Xu Z. Absolute measurement of roughness and lateral-correlation length of random surfaces by use of the simplified model of image-speckle contrast. Applied Optics 2002;41:4148–56. [12] Leonard LC. Roughness measurement of metallic surfaces based on the laser speckle contrast method. Optics and Lasers in Engineering 1998;30: 433–40. [13] Hun C, Bruynooghea M, Caussignacb J.-M, Meyrueisa P. Study of the exploitation of speckle techniques for pavement surface. Proceedings of the SPIE 2006; 6341: 63412A. [14] Lukaszewski K, Rozniakowski K, Wojtatowicz TW. Laser examination of cast surface roughness. Optical Engineering 1993;40:1993–7. [15] Parry G. Speckle patterns in partially coherent light. In: Dainty JC, editor. Laser speckle and related phenomena. Berlin, New York: Springer-Verlag; 1984. p. 77–122. [16] Pedersen HM. On the contrast of polychromatic speckle patterns and its dependence on surface roughness. Journal of Modern Optics 1975;22: 15–24. [17] Sprague RA. Surface roughness measurement using white light speckle. Applied Optics 1972;11:2811–6. [18] Markhvida I, Tchvialeva L, Lee TK, Zeng H. Influence of geometry on polychromatic speckle contrast. Journal of Optical Society of America A: Optics, Image Science, and Vision 2007;24:93–7. [19] Ning YN, Grattan KTV, Palmer AW, Meggitt BT. Coherence length modulation of a multimode laser diode in a dual Michelson interferometer configuration. Applied Optics 1992;31:1322–7.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Surface roughness measurement by speckle contrast under the illumination of light with arbitrary spectral profile Lioudmila Tchvialeva a, Igor Markhvida a, Haishan Zeng a,b, David I. McLean a, Harvey Lui a,b, Tim K. Lee a,b,c, a
Photomedicine Institute, Department of Dermatology and Skin Science, University of British Columbia and Vancouver Coastal Health Research Institute, Vancouver, Canada V5Z 4E8 b Departments of Cancer Control Research and Cancer Imaging, BC Cancer Agency, Vancouver, Canada V5Z 1L3 c School of Computing Science, Simon Fraser University, Burnaby, Canada V5A 1S6
a r t i c l e in f o
a b s t r a c t
Article history: Received 28 November 2009 Accepted 12 March 2010 Available online 2 April 2010
Quantification of surface roughness greater than a micron is desirable for many industrial and biomedical applications. Polychromatic speckle contrast has been shown theoretically to be able to detect such roughness range using an appropriate light source with a Gaussian spectral shape. In this paper, we extend the theory to arbitrary spectral profile by formulating speckle contrast as a function of spectral profile, surface roughness, and the geometry of speckle formation. Under a far-field set-up, the formulation can be simplified and a calibration curve for contrast and roughness can be calculated. We demonstrated the technique using a blue diode laser with a set of 20 metal surface roughness standards in the range 1–73 mm, and found that the method worked well with both Gaussian and non-Gaussian surfaces. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Surface roughness Polychromatic speckle contrast Far-field Fraunhofer zone
1. Introduction Optical techniques for measuring surface roughness are advantageous for their noncontact and nondestructive properties. 3D mapping techniques that utilize scanning or image processing usually require complex hardware. On the other hand, techniques that measure only roughness statistics can be based on light scattering using low-cost, simple devices with rapid data acquisition times. The analysis of speckles that arise from coherent light scattered by a surface belongs to the latter. Although the theoretical foundations of speckle methods have been established few decades ago [1], this methodology has not yet been implemented into practical instrumentation due to technological constraints. Recent advances in light sources and registration devices have rekindled interest in speckle-based techniques. Using speckle for surface roughness can be classified in the following ways. One class of speckle techniques utilizes two or more speckle fields and requires a complex hardware design [2–4]. In general these techniques are not suitable for industrial applications or biological tissue examination. Another class overcomes this
Corresponding author at: BC Cancer Research Centre, Cancer Control Research Program, 675 West 10th Avenue, Vancouver, BC, Canada V5Z 1L3. Tel.: + 1 604 675 8053; fax: + 1 604 675 8180. E-mail address: [email protected] (T.K. Lee).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.03.004
disadvantage by analyzing simple speckle patterns of scattered light captured with a camera. This method can be further subclassified: analyzing either the (1) spatial [5–8] or (2) intensity [9–17] statistics for the speckle patterns. The main drawbacks of the spatial statistics approach are the difficulty of calibration and the lack of theoretical formulations, which result in qualitative ‘‘ad-hoc’’ roughness assessments. On the other hand, intensity statistics such as speckle contrast [9] is based on well-studied theories. Monochromatic speckle contrast has been successfully utilized for submicron range roughness measurements [10–12] for weak-scattering surface conditions where wavelength determines the physical upper limit for measurements. Monochromatic speckle has also been applied over a larger roughness range in two studies. However, such attempts yielded only qualitative results with a complex two-scale surface structure [13] in addition to some unexplained data that conflicted with the theory [14]. Polychromatic speckle contrast has been shown theoretically [15,16] for measuring roughness in ranges larger than a micron as long as an appropriate light source is used. Coherence length is the scale of measurement, and this fact was qualitatively demonstrated in [17]. The theoretical formulations in [15,16] rely on a Gaussian spectral shape, and the formulations to our best knowledge have not been realized in practice. In this paper, we describe how to measure roughness by polychromatic speckle contrast under illumination with an arbitrary spectral profile. In the theoretical part we present a formula for the numerical calculation of a calibration curve relating contrast with roughness. We also demonstrate a practical realization of the
ARTICLE IN PRESS L. Tchvialeva et al. / Optics and Lasers in Engineering 48 (2010) 774–778
775
technique using diode lasers that are inexpensive, reliable, and widely available. As an example we calculate the calibration curves separately for a red and a blue laser diode, each with its own multipeak, complex spectrum. We validate our technique using metal roughness standards in the range 1–80 mm. Though the set-up we used is very simple, we emphasize the design principles of the device for achieving consistent results.
2. Theory
Fig. 1. Influence of width of function on integral. The three Gaussian functions can represent any of the three bell-shaped functions.
Contrast C of any speckle pattern is defined as [9] C¼
sI /IS
,
ð1Þ
where /yS denotes an ensemble averaging, and sI is the standard deviation of light intensity I, with sI2 being the variance:
s2I ¼ /I2 S/IS2 :
ð2Þ
By illuminating a surface with a polychromatic light source that has a finite spectrum as well as a finite temporal coherence length, the intensity of polychromatic speckles is the sum of the monochromatic speckle pattern intensities: Z 1 FðkÞIðx,kÞdk, ð3Þ IðxÞ ¼ 0
where k is wave number, F(k) is the spectral line profile of the illuminating light, and x is a vector in the observation plane. Eq. (3) implies that the registration time is much greater than the time of coherence. In other words, speckle patterns created by individual wavelengths are incoherent and we summarize the intensity. The behavior of I(x) depends on many factors. In some areas of the observation plane, intensity I(x, k) for all k will have the same distribution and the contrast of the resultant speckle pattern will be same as the contrast of the single monochromatic pattern. In other areas, the patterns will be shifted and their sum produces a smoothed speckle pattern with a reduced contrast. The second moment of the intensity I(x) can be calculated using Eq. (3) to yield Z 1Z 1 Fðk1 ÞFðk2 Þ/Iðx,k1 ÞIðx,k2 ÞSdk1 dk2 : ð4Þ /I2 ðxÞS ¼ 0
0
Calculating the variance according to Eqs. (2)–(4) we obtain Z 1Z 1 s2I ¼ /I2 S/IS2 ¼ Fðk1 ÞFðk2 Þ½/Iðx,k1 ÞIðx,k2 ÞS 0
0
/Iðx,k1 ÞS/Iðx,k2 ÞS dk1 dk2
ð5Þ
It has been shown in [18] that Eqs. (3) and (5) can be transformed to Z 1 Z 1 Z s2I ¼ Fðk1 ÞFðk1 þ DkÞdk1 jMDh ðDkÞ2 j exp½iDkRdSj2 dDk: 0
1
ð6Þ /IðxÞS ¼ S
Z
1
FðkÞdk,
ð7Þ
of intensity depending on the light source spectrum F(k), the second term R(Dk) relates to surface roughness, and the third term D(Dk)¼|R exp[ iDkR]dS|2/S2 represents the geometry of speckle formation. Eq. (8) has an important methodological meaning: it shows that contrast depends on three factors, namely, light coherence, surface roughness, and geometry. A group of techniques based on contrast measurement can be developed to determine any of these factors. As a rule, functions G(Dk), R(Dk), and D(Dk) have a bell-shape form (or limited width). To simplify the development of a measurement technique, relationships among the scale of coherence length, roughness, and optical paths can be chosen in such a way to allow the exclusion of unnecessary dependencies in Eq. (8) and/or varying the sensitivity of the contrast measurement technique to the appropriate parameter. This concept is illustrated in Fig. 1 where three bell-shaped functions are shown and an integral (contrast) depends mainly on width of function 3. Eq. (8) can be simplified for the purpose of roughness measurement. We will use only the central part of the speckle pattern x E0 to obtain the relationship between contrast and surface roughness. If we configure an optical set-up in a way that the illuminated surface area is smaller than coherence zone [9,18], the third integral in Eq. (8) approaches S2. For example, this condition will be satisfied when a parallel beam illuminates a spot q, which is in a distance z away from the camera, and Dkq2/z51. This is true for free geometry speckle in the area of Fraunhofer diffraction (zFraunhofer) and even for a distance z which is n times smaller than zFraunhofer, where n ¼lc/l, lc is a coherence length, and l is the average wavelength for the spectrum. If the surface height deviation has a normal distribution with a root mean square (rms) height deviation sh, the characteristics function of the surface height becomes [9] 2
MDh ðDkÞ ¼ eðð2sh Þ
Dk2 =2Þ
:
ð9Þ
Finally, combining Eq. (1) and Eqs. (6)–(8), the polychromatic speckle contrast can be expressed as R1 R1 2 0 ð 0 FðkÞFðk þ DkÞdkÞexpðð2sh DkÞ2 ÞdDk : ð10Þ C 2 ðsh Þ ¼ R1 ð 0 FðkÞdkÞ2 Knowing the emission light source spectrum F(k) and performing a simple numerical calculation of Eq. (10), the calibration curve for the contrast C vs. the RMS roughness sh can be obtained.
0
where Dk¼k2 k1, R is the full geometrical path of elementary waves from the source to the observation plane (excluding the additional phase shift by the diffuser), S is the illuminated spot area, and MDh ðDkÞ is the characteristic function of the surface height. Substituting Eqs. (6) and (7) into Eq. (1) results in Z 1 GðDkÞRðDkÞDðDkÞdDk: ð8Þ C2 ¼ 1
R 1 In Eq. (8) the first term GðDkÞ ¼ 0 Fðk1 ÞFðk1 þ DkÞdk1 = 2 R 1 inside the integral is the autocorrelation function 0 FðkÞdk
3. Calibration curves for blue and red diode lasers One of the most commonly used low-coherent light sources is the diode laser. A typical diode laser resonator is a Fabry–Perot interferometer with multi-peak emission spectrum [19]. In Fig. 2, spectra of two diode lasers are shown: (a) a blue diode laser (20 mW, BWB-405-20E, B&W Tek, Inc., USA) and (b) a red fiber-coupled diode laser (5 mW, 57PNL054/P4/SP, Melles Griot Inc., USA). These spectra were obtained by a Bomem DA8 Fourier transform interferometer system, with a quartz (visible) beamsplitter and a Si diode detector.
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To find a calibration curve we performed a numerical integration of Eq. (10) with the experimental diode laser spectra converted to F(k). The calculated dependence of contrast vs roughness is shown in Fig. 3(a). It should be noted that these curves do not describe contrast behavior in the submicron range. For comparison, we include the calibration curve for a light source with a Gaussian profile and with the same halfwidth as the red laser spectrum envelope in Fig. 3a (dotted line). Despite the similar trend for calibration curves generated by the Gaussian [15,16] and non-Gaussian sources, it is evident that the former curve cannot be used for precise measurements. For example, a speckle contrast of 0.8 corresponds to roughness about 60 mm using the curve calculated from (10) while the Gaussian spectrum example results in a roughness value of about 150 mm. In Fig. 3(b), we show the first derivates of the normalized calibration curves vs. roughness to find the highest measurement sensitivity and to identify the effective measurable roughness range for our light sources. The blue diode works better for roughness of around 20 mm, while for the red diode it is about 40 mm. The accuracy of the contrast measurements that we obtained in experiments was approximately 0.01. Using the first derivative we can estimate that the best accuracy we can expect is about 1 mm for the blue laser and about 2 mm for the red. The blue diode provides better accuracy overall, and both lasers can be effectively used for roughness ranging up to 100 mm.
For the experimental targets, we chose a set of 20 metal surface roughness standards with three types of finishing: gridblasted (G-6, Gar Electroforming Division, Electroforming, Inc., USA), spark erosion (Microshurf #331, Rubert+ Company Ltd., UK), and casting (Microshurf #334, Rubert +Company Ltd., UK). We scanned these surfaces with a WYKO profilometer and calculated their rms roughness. The set of 20 surfaces provided a roughness range 1–73 mm. Analyzing the histograms of roughness showed that most of the surfaces had a Gaussian distribution. Some exceptions were noted for the large scales; Fig. 4 illustrates the most significant difference. While Microshuft #331 had a Gaussian distribution, the Microshuft #334 histogram deviated from a Gaussian function significantly. Therefore, we were able to test our method with both Gaussian and nonGaussian surfaces. The laser speckle set-up is shown in Fig. 5. A parallel laser beam illuminated a surface of interest. The scattered light was acquired by a monochrome CCD camera (mvBlueFOX-M124G, Matrix Vision GmbH, Germany) without an imaging lens. The Pellicle beam splitter provided a 01 beam incidence and speckle pattern observation in the specular direction. Although the optical configuration is very simple, the principles described next must be observed in order to achieve proper measurements. The key design points are proper level of intensity, well-placed distances amongst the source, target and the camera, appropriate speckle size, and reduction of camera dark current and ambient illumination. We discovered that the average intensity can be estimated as /IS¼2m/ln N, where m is the depth of a camera pixel and N is the number of speckles. The derivation of this equation is out of the scope of this paper and we are preparing another article to fully analyze the effect of the camera resolution on the average intensity. Essentially, the typical value of /IS for a 8-bit camera
4. Experiment
Intensity (arb. units)
Intensity (arb. units)
In this section, we discuss the set-up of the experiment and compare the experimental results with the calculated calibration curves.
407.1 407.3 407.5 407.7 407.9 408.1
657.5
658
Wavelength, nm
658.5
659
659.5
Wavelength, nm
1
0.008
0.8
0.006
Slope, 1/µm
Speckle Contrast
Fig. 2. Actual diode lasers spectral profiles: (a) blue diode laser and (b) red diode laser. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0.6 0.4 0.2
0.004 0.002 0.000
0 0
50 100 150 Roughness, µm
200
0
50
100 150 Roughness, µm
200
Fig. 3. (a) Calibration curves and (b) their first derivatives, which show the sensitivity of the corresponding calibration curve. Solid and dash lines represent the calibration curves for the red and blue lasers, respectively. Dotted line on (a) corresponds to the calibration curve generated by a Gaussian source. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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777
1.1
8
1.0
MS 331 MS 334 Gaussian
6
0.9 0.8 contrast
0.7
4
0.6
Red theory Blue theory 331 Red 334 Red G6 Red 331 Blue 334 Blue G6 Blue
0.5 0.4
2
0.3 0.2
0 -0.25
-0.15
-0.05 0.05 Height, µm
0.15
0.25
0.1 0.0
Fig. 4. Comparing the probability density functions of the roughness for Microshuft #331(sh ¼ 62.5 mm) and Microshuft #334(sh ¼ 62.5 mm) with a Gaussian function(sh ¼ 62.5 mm).
0
20
40
60
roughness, µm Fig. 6. Comparison of calibration curves with experimental validation.
CCD Camera Polarizer
Beam splitter laser
Collimator
ensemble averaging can be replaced by spatial averaging which in turn means that we can use data from each pixel to calculate the average intensity and its standard deviation. The experimental procedure described in previous paragraphs was applied to a set of 20 metal standards. The measured contrasts are plotted against the rms roughness of the standards in Fig. 6. A good agreement between theory and experiment is observed for each surface finishing type. As expected, the smooth surfaces with roughness less than 8 mm form speckle patterns with contrasts close to unity.
5. Conclusion Rough surface Fig. 5. The speckle geometry set-up.
should be around 30. For metal standards, it is a good idea to employ a polarizer to prevent overexposure. The camera–surface distance z and the beam width q determine the size of speckles s which should be in the order of lz/q. Reducing the speckle size leads to the reduction of the contrast because of spatial averaging effect over the camera pixels. Similarly, a small amount of over-sized speckle also causes lower accuracy in contrast measurement. The minimum speckle should be at least twice the pixel size; speckle size is typically adjusted to be about 5–10 CCD camera pixels which can be easily controlled by analyzing the image. The surface to camera distance z should be large enough for the far-field (Fraunhofer) zone. As mentioned earlier it can be decreased n ¼lc/l times if necessary. In our experiment the optimal distance z was 270 mm; the illuminated spot diameters were 3 mm for the blue laser and 4.5 mm for the red laser. These settings provided the average speckle size of 60 and 80 mm for the blue and red, respectively. Camera dark current and ambient illumination can be removed if we subtracted an image without laser exposure from the speckle images. It should be noted that if the above conditions are achieved, the speckle pattern will be spatially uniform and
Using inexpensive diode lasers allows one to determine the surface roughness in the tens of micron range (10–100 mm) by simply measuring polychromatic speckle contrast. It was shown that models based on Gaussian spectra can be used only for qualitative estimations. For precise measurements the calibration curve for contrast should be calculated by Eq. (10). Despite the assumption of Gaussian height distribution used in the model, our experiments showed that the technique can be applied for a wide class of surfaces characterized by both Gaussian and non-Gaussian height distributions. We believe that this study contributes to the industrial application of speckle contrast, and will be of interest for surface metrology applications.
Acknowledgements This work was supported in part by Grants from the Canadian Institutes of Health Research, the Natural Sciences and Engineering Research Council of Canada, Canadian Dermatology Foundation, UBC & VGH Hospital Foundation, and UBC Faculty of Medicine. Also we would like to thank Dr. M.L. Thewalt (Simon Fraser University, Burnaby, Canada) for measuring the emission spectra of the red and the blue diode lasers and Mr. S. Yick (Institute for Fuel Cell Innovation, Vancouver, Canada) for measuring the target surface roughness.
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