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We present a coherence scanning interferometer configured to deal with rough glass surfaces exhibiting very low reflectance due to severe sub-surface light scattering. A compound light source is prepared by combining a superluminescent light-emitting diode with an ytterbium-doped fiber amplifier. The light source is attuned to offer a short temporal coherence length of 15 μm but with high spatial coherence to secure an adequate correlogram contrast by delivering strongly unbalanced optical power to the low reflectance target. In addition, the infrared spectral range of the light source is shifted close to the visible side at a 1,038 nm center wavelength, so a digital camera of multi-mega pixels available for industrial machine vision can be used to improve the correlogram contrast further with better lateral image resolutions. Experimental results obtained from a ground Zerodur mirror of 200 mm aperture size and 0.9 μm rms roughness are discussed to validate the proposed interferometer system.
© 2017 Optical Society of America
1. Introduction
Very low thermal expansion glass is commonly used as substrates for telescope mirrors in both space and terrestrial applications [1,2]. Such glass is machined into a desired convex or concave surface usually in two steps; first by coarse grinding and then by fine polishing. The whole machining is a time consuming process in which the glass surface needs to be checked from time to time until it meets all the precision requirements as a final product. Optical interferometric techniques such as Fizeau or Twyman-Green interferometers are widely adopted for the testing of mirrors using visible or near-infrared light [3]. However, unlike metals, when the glass surface is in a rough state in the middle of grinding ahead of polishing, the reflectance from the glass surface is too weak to be treated by the conventional interferometers. Adopting long-wavelength infrared light such as CO2 laser may mitigate the problem [4], but the lateral measurement resolution has to be sacrificed due to the lack of high pixel resolution cameras operating in the non-visible wavelength range. Thus, during early stage of machining, contact-type profilometry is preferably used at the cost of inspection and ultimately total machining time [5].
Traditionally, rough surfaces have been measured using low temporal coherence light by means of scanning interferometry [6–9]. This method permits 3-D surface reconstruction by point-by-point processing of individual coherence scanning correlograms, thereby avoiding the 2π-phase ambiguity that would be encountered in the case of high coherence laser interferometry during spatial phase unwrapping. In order for the coherence scanning interferometric method to be applicable for rough ground glass, the low temporal coherence light delivered to the target surface should be aligned in a collimated or diverging wave with strong optical power to cope with extremely low reflectance. Conventional white-light sources or light-emitting diodes are found not suitable for generating such a high quality strong wave particularly for large-aperture surfaces. The main reason is the lack of spatial coherence, so the source optical power is lost mostly in the process of beam collimation or diverging due to poor focusing capability.
In this study, an attempt is made to devise a new light source of scanning interferometry that is capable of providing low temporal coherence but with high spatial coherence. Specifically, a superluminescent light-emitting diode (SLD) is combined with an ytterbium-doped fiber amplifier (YDFA) to provide a ~0.1 W optical power with high spatial coherence through a single-mode fiber exit of 6 μm core diameter. The temporal coherence of the light source is adjusted to ~15 μm so as to obtain optimized correlograms from rough ground surfaces. In addition, the wavelength range is centered at 1,038 nm so that an ordinary machine vision digital camera operating in the visible to near infrared range with mega pixels can be used to improve the correlogram contrast with a fine lateral imaging resolution. Experimental results obtained from a ground Zerodur mirror of 200 mm size and 0.9 μm rms roughness are discussed to validate the proposed light source and interferometer system.
2. Coherence scanning interferometer
When a rough surface is illuminated by a light source in two-arm interferometry, the specular reflection that constitutes the measurement wave reduces drastically due to scattering as the surface roughness increases. Assuming the height distribution of the surface to be Gaussian, the specular reflectance R is theoretically derived as R = R0 exp[-(4πσh/λ)2]; R0 denotes the reference reflectance for the perfectly smooth surface of the same material, and λ is the wavelength of the light source and σh is the rms surface roughness [10,11]. For rough glass, the reference reflectance R0 has a small value of about 0.04, and severe light scattering due to sub-surface cracks makes the specular reflectance R be usually two or more orders weaker compared to metal surfaces of same roughness. This fact requires special care to secure the correlogram contrast adequately from rough glass by strengthening the measurement wave against the reference wave without loss of the beam coherence quality.
Figure 1(a) shows the light source configured in this study to deal with rough glass surfaces by coherence scanning interferometry. The light source is based on a superluminiscent light-emitting diode (SLD) offering an 8 mW output power over a 38 nm spectral bandwidth about a 1,045 nm center wavelength [12]. An ytterbium-doped fiber amplifier (YDFA) is connected in series to increase the output power to 97.5 mW, of which the output beam is made linearly polarized. At the same time, the gain narrowing effect of the fiber amplifier [13] reduces the spectral bandwidth to 18.9 nm about a center wavelength of 1,038 nm as shown in Fig. 1(b). Another merit of adopting the fiber amplifier is to enhance the spatial coherence through a single-mode fiber exit of 6 μm core diameter so that the output beam can be extended to a large diameter without loss of the optical power. Further, the compound light source can be detected with reasonable sensitivity by ordinary digital cameras of multi-mage pixels used for industrial machine vision in the wavelength range of 400 – 1,100 nm [14].
Fig. 1 Compound light source constructed for coherent scanning interferometry of rough glass surfaces. (a) Optical configuration of the light source based on a superluminiscent light-emitting diode (SLD). (b) Optical spectrum at the exit fiber of the ytterbium-doped fiber amplifier (YDFA). (c) Coherent correlogram obtained from a mirror target surface to measure the temporal coherence length.
Compared to visible light, the compound infrared source constructed in this study suffers a loss in camera sensitivity but offers advantage of increasing the specular reflectance from rough surfaces with longer wavelengths. In addition, the step size to implement coherence scanning interferometry can be made larger so as to reduce the total measurement time including computation. Figure 1(c) shows a typical pattern of coherence correlogram obtained using the infrared light source by scanning the reference mirror with 100 nm steps on a Twyman-Green interferometer for a smooth target surface. With a target surface being made of the same material as the reference mirror, the correlogram contrast is monitored to be as high as 0.9. The temporal coherence length lc is measured to be 15.1 μm, which well agrees with the analytical value estimated from the spectral bandwidth measured in Fig. 1(b). For successful implementation of coherence scanning interferometry, the temporal coherence length is required to be at least four times larger than the standard deviation of the surface height variations [15]. This implies that the light source is able to treat ground rough surfaces with rms roughness as large as 3.8 μm.
The light source was tested by measuring a ground surface fabricated on a Zerodur glass mirror of 200 mm diameter with an rms roughness of 0.91 μm. The test surface was of spherical shape with a ~1,013 mm radius of curvature. Figure 2 shows the interferometer system configured in a Twyman-Green type for the test measurement with several noteworthy features. First, the high spatial coherence of the source light permits to produce a well-collimated beam of 20 mm diameter without significant loss of the source power. Second, in order to protect the extremely low coherent reflection from the test mirror, an angled parabolic metal mirror is used for illumination without backward reflection. In addition, no lenses are used in the measurement arm to minimize stray light reflection from lens surfaces. Third, in order to deliver as much light as possible to the measurement wave, polarization-based beam splitting is made using a half-wave plate (HWP) in combination with a quarter-wave plate (QWP). The wave plates are slightly tilted to prevent their backward reflection rays from reaching the camera. Fourth, the optical power of the reference wave is controlled precisely by adjusting the linear polarizer (LP) installed in front of the camera imaging lens. Finally, it is conformed that an extremely unbalanced splitting ratio of ~106 is achieved between the measurement wave and the reference wave.
Fig. 2 Interferometer configuration for testing a large rough glass surface. CL: collimating lens, HWP: half-wave plate, QWP: quarter-wave plate, PBS: polarizing beam splitter, LP: linear polarizer, IL: imaging lens.
For implementation of coherence scanning interferometry, the reference mirror of the interferometer is moved using a PZT micro-actuator along the optical axis in a scanning mode. Resulting coherence correlograms are captured with a digital camera of complementary metal–oxide–semiconductor (CMOS) type that offers 3% sensitivity at 1,050 nm. The image plane of the camera is comprised of 2,560 × 2,160 pixels with a 6.5 μm pixel pitch, among which 1,200 × 1,200 pixels are allocated to image the test mirror surface. The imaging lens of the camera has a 4.6 f-number and a 0.039 lateral magnification. Detailed measurement conditions are summarized in Table 1. For a source power of ~0.1 W used in the experiment, the intensity of the measurement wave reaching the camera image plane is measured to be ~0.64 μW. This means that the test surface reflects only an amount of about 10−6 of the incident light, which is very small but detectable by the CMOS camera selected in this study.
Table 1. Coherence scanning interferometer conditions.
3. Measurement and discussion
The height of each point on the test surface is determined by applying the centroid algorithm [16], which identifies the weighted center location of each correlogram obtained by coherence scanning interferometry. The centroid algorithm allows the scan step to be selected as large as 200nm, within the Nyquist sampling limit of two data points per interference fringe, to minimize the total sampling time and computation. Figure 3(a) shows the z-map profile of the test surface reconstructed using the raw data without any post-processing. The height fluctuation of the measured surface profile turns out ~20 μm in peak to valley, which is rather larger than the actual height irregularities of the test surface measured by a stylus-type surface roughness measuring instrument. The discrepancy is speculated due to the speckle pattern generated by the rough test surface, which consequently affects the background intensity, correlogram contrast and measurement uncertainty of coherence scanning interferometry [17]. The measurement uncertainty δz is analytically estimated as δz = (1/√2)(<Is>/Is)1/2σh, in which Is is the speckle intensity at a surface point and <Is> is the mean of the speckle intensity over the entire field of view [18]. The analytical prediction implies that for a bright speckle background, the uncertainty becomes small and accordingly the measured height is subject to lesser error. On the other hand, for a dark speckle background, the signal-to-noise ratio of its correlogram weakens, thus the measurement uncertainty rises, in inverse proportion to the speckle intensity, by an order of magnitude comparably larger. Figure 3(b) shows a smoothened profile in which the speckle effect is suppressed using a median filter averaging 11 x 11 pixels, which leads to low-pass filtering with a spatial cut-off frequency of 0.55/mm.
Fig. 3 Measurement results. (a) Surface profile reconstructed using raw data. (b) Smoothened profile by median filtering with a spatial cut-off frequency of 0.55/mm.
For further insights of the speckle effect on the measurement uncertainty, Fig. 4(a) shows a plot depicting how 20 x 20 pixels in the central area of the measured data are correlated in terms of the speckle intensity and the correlogram contrast. The speckle intensity in the plot represents the normalized value with respect to the mean speckle intensity. The plot indicates there is a clear correlation that higher speckle intensities correspond to higher correlogram contrasts, improving the measurement accuracy. This tendency is extended in other parts of the measured surface.
Fig. 4 Comparative analysis results for correlogram contrast vs. speckle intensity in scanning coherence interferometry from rough glass.
It is also noted that correlograms sampled in the presence of speckles are sensitive to the actual pixel size of the camera in use. In other words, increasing of the pixel size generally leads to reduction of the correlogram contrast because more speckles are averaged within a single pixel [19]. This effect can be quantitatively analyzed in terms of N that is defined as the reciprocal pixel size with respect to the Airy disc diameter of the imaging system, i.e., N = Airy diameter / pixel size. The Airy disc diameter indicates the ultimate lateral resolution achievable in an imaging system, which is calculated to be 11.7 μm for the camera lens used for this experiment. Since the pixel size of the given camera is 6.5 μm, the value of N is calculated 1.79 when a correlogram is sampled from each pixel. For comparative analysis, N can be reduced by varying the effective pixel size by means of pixel binning, which is an electronic means of increasing the actual number of pixels allocated in producing a single correlogram. The result shown in Fig. 5 reveals that the correlogram contrast reduces for smaller N due to the increased overlap of multiple speckles. In other words, the effective pixel size needs to be as small as possible for better measurement accuracy, which is definitely an advantage of adopting the visible camera of five mega pixels, instead of infrared cameras of limited pixels, as attempted in this study. Figure 5 also shows that the probability density function of the correlogram contrast follows a gamma distribution, regardless of N, in the same way as the speckle intensity predicted by theory of speckle statistics [20].
Fig. 5 Histogram of correlogram contrast for various pixel sizes; (a) N = 1.79, (b) N = 0.90, (c) N = 0.45 and (d) combined plot in terms of the probability density.
Finally, the measured profile of Fig. 3(b) obtained from the proposed coherence scanning interferometer system was post-processed so as to remove the interferometer system error by means of data averaging with three rotations [21]. In addition, the piston, tilt and defocus error terms were eliminated by adopting the Zernike polynomial decomposition [3,22]. Then the measurement result was compared with another result obtained using a contact-type profiler (UA3P-5) operating on atomic force microscopy [23]. The latter was measured from the same test surface by setting up the profiler to offer a profile accuracy of 0.1 μm over a 200 mm x 200 mm work area. Figure 6 compares two measurement results and Table 2 summarizes comparative results, in which the rms difference between the two measurement results is found less than a micrometer.
Fig. 6 Final measurement results of the test surface: measurement results obtained using (a) the proposed method and (b) contact-type profiler and (c) the difference between two measurements.
Table 2. Comparison of Measurement Results.
4. Conclusion
We have demonstrated a coherence scanning optical interferometer for profile measurement of large-area rough glass surfaces of extremely low reflectance. A compound light source was devised by combining a superluminiscent light-emitting diode with an ytterbium-doped fiber amplifier. The light source was used to illuminate the target surface with a ~0.1 W optical power with high spatial coherence through a 6 μm core-diameter single-mode fiber. With the spectral range of the light source being centered at a 1,038 nm wavelength with a 19 nm bandwidth, coherence scanning interferometry was performed with a 15 µm coherence length using a standard machine vision digital camera of four mega pixels. The test measurement showed that the entire surface of a Zerodur glass of 200 mm diameter with a 0.9 µm rms roughness can be measured with a sub-micron discrepancy in comparison with a contact-type profiler. With a short measurement time of ~5 min, the proposed interferometric method may be applicable to various large-area rough surfaces of very low reflectance.
Funding
National Research Foundation of South Korea (NRF-2012R1A3A1050386).
Acknowledgments
The authors appreciate J. You of Lasernics Inc. for supports in preparing the light source used in the experiment.
References and links
1. P. R. Yoder, Opto-Mechanical Systems Design, 3rd ed. (CRC, 2006). Chap. 3.
2. P. Y. Bely, The Design and Construction of Large Optical Telescopes (Springer, 2002), Chap. 4.
3. D. Malacara, Optical Shop Testing, 2nd ed. (John Wiley & Sons, 1992).
4. O. Kwon, J. C. Wyant, and C. R. Hayslett, “Rough surface interferometry at 10.6 microm,” Appl. Opt. 19(11), 1862–1869 (1980). [CrossRef] [PubMed]
5. Y. Wang, P. Su, R. E. Parks, C. J. Oh, and J. H. Burge, “Swing arm optical coordinate-measuring machine: High precision measuring ground aspheric surfaces using a laser triangulation probe,” Opt. Eng. 51(7), 073603 (2012). [CrossRef]
6. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31(7), 919–925 (1992). [CrossRef] [PubMed]
7. P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015). [CrossRef]
8. J. Schmit, “White-Light Interference 3D Microscopes,” in Handbook of Optical Dimensional Metrology, K. Harding ed. (Taylor & Francis, 2013), pp. 395–418.
9. X. Colonna de Lega, J. Biegen, P. de Groot, G. Häusler, and P. Andretzky, “Large field-of-view scanning white-light interferometers,” in Annual Meeting of the American Society for Precision Engineering, Proc. ASPE 1275, (2003).
10. H. E. Bennett and J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51(2), 123–129 (1961). [CrossRef]
11. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53(12), 1389–1394 (1963). [CrossRef]
12. C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46(12), 1763–1774 (1999). [CrossRef]
13. Y. Chiba, H. Takada, K. Torizuka, and K. Misawa, “65-fs Yb-doped fiber laser system with gain-narrowing compensation,” Opt. Express 23(5), 6809–6814 (2015). [CrossRef] [PubMed]
14. P. Magnan, “Detection of visible photons in CCD and CMOS: A comparative view,” Nucl. Instrum. Methods Phys. Res. A 504(1-3), 199–212 (2003). [CrossRef]
15. P. Pavliček and J. Soubusta, “Theoretical measurement uncertainty of white-light interferometry on rough surfaces,” Appl. Opt. 42(10), 1809–1813 (2003). [CrossRef] [PubMed]
16. S. Chen, A. W. Palmer, K. T. V. Grattan, and B. T. Meggitt, “Digital signal-processing techniques for electronically scanned optical-fiber white-light interferometry,” Appl. Opt. 31(28), 6003–6010 (1992). [CrossRef] [PubMed]
17. P. Pavliček and O. Hýbl, “White-light interferometry on rough surfaces--measurement uncertainty caused by surface roughness,” Appl. Opt. 47(16), 2941–2949 (2008). [CrossRef] [PubMed]
18. B. Wiesner, O. Hybl, and G. Häusler, “Improved white-light interferometry on rough surfaces by statistically independent speckle patterns,” Appl. Opt. 51(6), 751–757 (2012). [CrossRef] [PubMed]
19. S. J. Kirkpatrick, D. D. Duncan, and E. M. Wells-Gray, “Detrimental effects of speckle-pixel size matching in laser speckle contrast imaging,” Opt. Lett. 33(24), 2886–2888 (2008). [CrossRef] [PubMed]
20. J. C. Dainty, “The statistics of speckle patterns,” in Progress in Optics 14, E. Wolf, ed. (Elsevier, 1977).
21. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 379–383 (1984). [CrossRef]
22. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]
23. S. Scheiding, C. Damm, W. Holota, T. Peschel, A. Gebhardt, S. Risse, and A. Tünnermann, “Ultra precisely manufactured mirror assemblies with well defined reference structures,” Proc. SPIE 7739, 773908 (2010). [CrossRef]
