High-resolution wide-dynamic range electronically scanned white-light interferometry Lazo M. Manojlović Zrenjanin Technical College, Đorđa Stratimirovića 23, 23000 Zrenjanin, Republic of Serbia ([email protected]) Received 10 December 2013; revised 17 March 2014; accepted 20 April 2014; posted 23 April 2014 (Doc. ID 202709); published 20 May 2014
A novel high-resolution wide-dynamic range electronically scanned white-light interferometry-based interrogation technique is presented. By using off-the-shelf optical components, this technique is capable of reaching a subnanometer resolution. The technique relies on a simple optical setup in which the wedge and camera axes are mutually inclined for a very small angle in the horizontal plane and two-dimensional fringe pattern analysis. Resolution below 0.3 nm and dynamic range of 106 dB have been achieved with a signal-to-noise ratio lower than 25 dB. © 2014 Optical Society of America OCIS codes: (060.2370) Fiber optics sensors; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.53.003341
1. Introduction
Electronically scanned white-light interferometry (WLI) represents a compact, rigid, stable, and fast interrogation technique since there are no mechanical moving parts present in the sensing systems employing this technique. However, this technique suffers from low resolution due to a low signal-tonoise ratio (SNR) and low sampling rate. Therefore, special attention should be paid to the signal processing in order to obtain subpixel resolution. Nevertheless, it is still difficult to reach subnanometer resolution with typical SNR values ranging from 20 to 40 dB [1,2]. A typical application of the electronically scanned WLI is in the signal conditioning of low-finesse fiber-optic Fabry–Perot interferometric sensors [3–6]. A resolution of 0.9 nm in the sensing interferometer path length difference measurement range of 30 μm has been achieved [6,7], thus providing a sensor dynamic range of 90.5 dB. However, in order to obtain this subnanometer resolution a birefringent wedge together with two linear polarizers and focusing optics has been employed. In this paper a novel high-resolution wide-dynamic range 1559-128X/14/153341-06$15.00/0 © 2014 Optical Society of America
electronically scanned WLI-based interrogation technique has been presented that is capable of reaching subnanometer resolution with a SNR lower than 25 dB. The resolution of 0.28 nm, accomplished by the simple setup only consisting of a wedge and camera, is pretty close to the theoretical minimum of 0.17 nm that is obtained for the same SNR [8]. Moreover, this simple configuration is capable of sensing interferometer path length difference measurement in the dynamic range of 106 dB. 2. Theory of Operation
A typical electronically scanned WLI-based sensing system is composed of a collimation lens, a wedge (Fizeau interferometer), and a linear photodetector array. A relatively low number of pixels of the linear array limits the sensor resolution [9]. There are two ways to increase the sensor resolution. The first is to decrease the wedge angle. However, in this way the sensor dynamic range is also decreased. The second way implies the use of linear arrays with a larger number of pixels. This further implies the use of long, custom-made, and expensive linear arrays and wedges that, on the other hand, due to thermal drift and ambient vibration, significantly deteriorate the sensor performances such as stability and accuracy. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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In comparison with a linear array a typical camera has an approximately three orders of magnitude larger number of pixels. Although the camera has a large number of pixels the main problem of involving cameras into the WLI measurement chain is their two-dimensionality. This problem can be overcome if the wedge and camera axes are mutually inclined in a horizontal plane for a small angle. Such a wedge and camera configuration gives the following wedge thickness tx; y above the camera: tx; y t0 x cos φ tan θ y sin φ tan θ ≈ t0 θx φθy;
(1)
where x and y are the coordinates in the camera plain, t0 is the wedge thickness at the coordinate system origin, φφ ≪ 1 is the angle between the wedge and camera axes, and θθ ≪ 1 is the wedge angle. Based on the camera captured two-dimensional lowcoherence interferogram (LCI), a novel algorithm has been developed aimed at finding the central fringe maximum (CFM) position with high resolution. The proposed algorithm consists of two steps. The first step is based on the camera row analysis. Therefore, it is necessary to extract a LCI from a single row, e.g., from a row with the maximal fringe visibility. The wedge thickness along this row positioned at y y0 is given by tx; y0 ≈ t0 φθy0 θx:
(2)
By using one among many algorithms for analyzing LCI [9] one can obtain an estimation of the CFM position. Due to the low SNR and low sampling rate, this estimation is rather poor. Nevertheless, the efficient algorithm will identify at least the position close to the central fringe. When the position, i.e., the pixel in the analyzed row that is in close proximity to the CFM, has been identified, we switch on the second step. In the second step of the algorithm, the camera column that includes this pixel is to be analyzed. Along this column the wedge thickness is given by txCF ; y ≈ t0 θxCF φθy;
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tCFM ≈ t0 θxCF φθyCFM :
(4)
This wedge thickness directly corresponds to the optical path difference (OPD) of the WLI sensing interferometer. Typically, the WLI sensing interferometer OPD is twice as much as the corresponding wedge optical thickness. According to Eq. (4) the single pixel resolution in the measuring WLI sensing interferometer OPD is given by δL ≈ 2npφθ, where n is the wedge index of refraction and p is the distance between the centers of the adjacent camera pixels. In order to get better insight into all relevant geometrical parameters of the wedge and camera setup, in Fig. 1 we have presented the top view of the setup, together with both side views of the setup cross sections along both relevant axes. In Fig. 1, we have also presented the active zone of the camera that represents the rectangle with maximal dimensions on the camera surfaces that is covered by the wedge. Only the images captured by this active zone will be taken into consideration for further signal processing. 3. Experimental Setup
In order to test the capabilities of the proposed WLI interrogation technique a simple low-coherence fiber-optic absolute position sensor has been built. The layout of the experimental setup, based on a single-mode 2 × 2 fiber-optic coupler (FOC), is presented in Fig. 2. The sensing interferometer is a lowfinesse Fabry–Perot interferometer that is formed between the end of the sensing fiber (SF), thus forming the reference mirror (RM) and sensor mirror (SM). To avoid parasitic interference, the distance D between the RM and SM, i.e., the cavity length, must be greater than one half of the used white-light source (WLS) coherence length LC D > LC ∕2. As the WLS a pigtailed superluminescent diode (SLD) is used, which is driven by the commercial current and temperature controller (CTC). SLD emits infrared light with a central wavelength of 842 nm and
(3)
where xCF represents the position of the pixel in close proximity to the central fringe that has been identified in the previous step. If we compare Eqs. (2) and (3) we can notice that wedge thickness changes very slowly along the column providing us an equivalent wedge with a much smaller angle than in the first step. Therefore, a much higher pixel resolution in the CFM positioning can be achieved. In order to reach maximal resolution, angle φ should be chosen as small as possible but still large enough so that each column covers just a couple of fringes. After finding the pixel position yCFM along the observed column that is closest to the CFM that is further closest to the position y y0 of the pixel identified in the 3342
first algorithm step, the wedge thickness tCFM above this particular pixel is given by
Fig. 1. Top view of the setup, together with both side views of the setup cross sections along both relevant axes.
Fig. 2. Layout of the experimental setup.
with a full-width at half-maximum spectral bandwidth of 22 nm. The total optical power launched by the SLD in a single-mode fiber is approximately 1.4 mW. The inactive FOC arm is immersed into the index matching gel (IMG) in order to maximize the interferogram fringe visibility. Finally, the backreflected light is used for wedge (W) illumination. The LCI is captured by the camera (C) and transferred to the personnel computer (PC) for further processing. 4. Results and Discussion
The camera captured images for three distances between RM and SM: (1) D 362 μm, (2) D 370 μm, and (3) D 392 μm are presented in Fig. 3. These distances are within the range in which the highest fringe visibilities of the sensing interferometer have been observed. These two-dimensional LCIs are obtained for φ ≈ 1.84° and θ 0.5°. In order to provide the highest possible level of the interferometric signal, both sides of the wedge are partially reflective with 30% of reflectivity [10]. In Fig. 3, one can clearly notice the wedge edge in the bottom part of each image. Since there are not any coupling optics the wedge illuminating FOC fiber is positioned at a relatively large distance from the wedge surface in order to provide roughly uniform illumination of the camera surface. Moreover, such an arrangement also provides a relatively high spatial coherence of the light beam impinging onto the camera. According to the first step of the proposed algorithm a single row of the captured image should be extracted for further processing. To be able to extract the pixel position/number closest to the CFM the raw data from a single row that are depicted in
Fig. 3. Camera captured two-dimensional LCIs.
Fig. 4(a) should be further processed. The typical signal processing chain of the electronically scanned WLI signal consists of bandpass filtration, normalization with low-pass filtered signal component [11], and envelope detection and fitting. All these signal processing steps are depicted in Figs. 4(b)–4(d), respectively, where lines 1 (circles), 2 (squares), and 3 (triangles) mark the corresponding signals for the following RM–SM distances: D 362 μm,
Fig. 4. Signal processing chain for three RM–SM distances. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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D 370 μm, and D 392 μm, respectively. The normalization step has been performed in order to suppress the influence of the nonuniform irradiance distribution over the camera surface. The typical SLD has Gaussian spectral distribution [12]. Therefore, the Gaussian function has been used for envelope fitting in the last step. Based on the fitting parameters, where the goodness of fit parameter R2 was greater than 0.98 in all three cases, it is possible to estimate the pixel position/number closest to the CFM position. According to Eq. (3), the corresponding wedge index of refraction n 1.51 and distance between the centers of the adjacent camera pixels p ≈ 2.2 μm, and taking into consideration that WLI sensing interferometer cavity length is equal to the corresponding wedge optical thickness, the single pixel resolution of CFM position estimation in the first algorithm step is equal to ΔD ≈ npθ ≈ 29 nm. This relatively low resolution is still sufficient for identifying the central fringe. After identifying the corresponding pixel number in the first algorithm step we switch on the second step. In order to eliminate the influence of the nonuniform irradiance distribution, bandpass filtration and normalization have been performed over each image row. The rows in the bottom part of the images depicted in Fig. 3, which contain pixels in close proximity to the wedge edge, are omitted from further processing. When filtration and normalization have been performed the column signal that contains the aforementioned pixel should be of the cosine shape. The exact position of the CFM is obtained according to the above presented algorithm in the introductory part of this paper. Based on the values of the experimental setup parameters the estimated overall single pixel resolution in CFM position measurement is δD δL∕2 ≈ 0.93 nm. The overall performances of the proposed algorithm are estimated based on a simple experiment. In the presented experimental setup, where SM was positioned at a distance of approximately 370 μm from RM, SM was slightly shifted with the help of a calibrated high-precision piezo actuator. Both captured images, obtained for the initial and shifted SM positions, were processed according to the presented algorithm. The results, when the SM shift was 5 nm, are presented in Fig. 5. As the result of the first step of the algorithm the Gaussian fitting parameters are obtained, where envelope data and Gaussian fits are presented in Figs. 5(a) and 5(b) for 1 (circle) initial and 2 (square) shifted SM position. Figure 5(b) represents the enlarged view of Fig. 5(a) in close proximity to the maximum value of the fitted function. According to the fitting parameters CFM is positioned with a standard deviation of approximately 1.7 pixels, i.e., with a resolution of approximately 48 nm, where the SNR of the row signal was 24.7 dB. This is still sufficient for central fringe identification, but insufficient to detect the shift of 5 nm. Moreover, the measured SM shift is in the opposite direction from the real one. However, after 3344
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Fig. 5. Measurement process for the SM small shift estimation.
rounding both of the corresponding fitting parameters refer to the same pixel. After identifying the pixel in the inspected row closest to the CFM we switch on the second algorithm step. The corresponding column signals are depicted in Fig. 5(c). One can notice that both signals are overlapping due to the very small SM shift. Therefore, an enlarged view of the Fig. 5(c) central part is presented in Fig. 5(d) in close proximity to the CFM, where lines 1 and 2 represent the initial and shifted SM positions, respectively, and lines 3 and 4 represent their corresponding fitting curves of the cosine shape, where the goodness of fit parameter R2 was greater than 0.987 in both cases. Instead of CFM the central fringe minimum value has been presented in Fig. 5(d). The reason for this lies in the fact that RM and SM have different phase shifts, 0 and π, respectively, after the light beam reflection. From Fig. 5(d) it is evident that the column signal shifts toward the first column pixel. This can be confusing if compared to the row shift obtained in the first algorithm step. However, this is expected as the wedge thickness decreases along the inspected column from the top to the bottom and the first pixel is located at the top of the captured image. Finally, according to the fitted parameters the measured shift is 5.5 nm and its standard deviation is 0.28 nm (0.31 pixels), which at the same time represents the overall sensing system subpixel resolution. In order to estimate the dynamic range of such a built sensing system it will be assumed that the maximally measured cavity length range corresponds to the maximal wedge optical thickness difference along the camera surface. So, for the maximal RM–SM distance range DM we have DM ≈ npN X θ ≈ 59 μm, where N X is the row number of pixels, for which for the used camera we have N X 2048. The dynamic range is defined as the
ratio of the measurement range and resolution. Since SNR changes only for 1.1 dB when the RM–SM distance changes in the range from 362 to 392 μm the resolution of 0.28 nm, which was estimated for the distance of 370 μm, will be taken as the relevant one for the whole measurement range. In this respect the dynamic range is approximately 106 dB. This relatively high dynamic range can be increased if the angle between the wedge and camera decreases. However, there is a limit in decreasing of this angle. In order to cover the entire measurement range with the highest resolution, each column must cover at least the range of a single row pixel; i.e., N Y δD ≥ ΔD must be fulfilled, where N Y is the column number of pixels. In our case we have N Y 1536, where φ ≥ 1∕N Y ≈ 0.037° is valid. Consequently, the maximally theoretically achievable dynamic range of such a composed sensing system is given by 20 log10 DM ∕δD ≈ 20 log10 N X N Y ≈ 130 dB, where the single column pixel resolution has been considered as the relevant resolution of the complete measurement setup. If compared with the single pixel dynamic range of the proposed sensing system of 96 dB, where the achieved single pixel resolution of δD ≈ 0.93 nm has been taken into account, one can notice that there is a space in the sensing system performance improvement. In practice this is difficult to achieve as the angle between the wedge and the camera must be very small, thus reaching higher noise levels and consequently lower resolution. This can be easily seen in Fig. 6(a), where we have presented the power spectral density of the analyzed raw column signal (arrow marks the corresponding carrier signal). If the angle between the wedge and the camera decreases the carrier signal will move toward the lower frequencies where we have higher noise levels due to the flicker noise. Figure 6(b) also presents the power spectral density of the analyzed raw row signal. One can clearly notice the carrier signal marked with an arrow. If the angle between the wedge and the camera is kept very small (φ ≪ 1) there will be no significant change in the carrier signal position. The carrier signal position will be changed only if we chose the wedge with a different angle. Nevertheless, one can notice that the wedge angle has been chosen appropriately as
Fig. 6. Power spectral densities (PSD) of the raw (a) column and (b) row signal.
the carrier signal is positioned away from the high level of flicker noise. In the end, the capability of the first algorithm step to identify the pixel in close proximity to the CFM has been tested. Similarly to the previous test the SM was positioned at a distance of approximately 370 μm from the RM. Since the 568th row of the captured images has been identified as the one with the highest fringe visibility the first algorithm step has been performed over this row. As a result of the first step the 923rd pixel is identified to be the closest one to the CFM. After switching to the second step and analysis of the 923rd column, the 522nd pixel in the inspected column has been identified as the one closest to the CFM. Therefore, an error of 46 column pixels, i.e., an error of 43 nm, occurred in the first algorithm step. This measured error is very close to the previously estimated resolution of the first algorithm step of approximately 48 nm. 5. Conclusion
In conclusion, a novel high-resolution wide-dynamic range electronically scanned WLI-based sensing system is presented in this paper. The subnanometer resolution has been reached by small angular shift of the wedge with regard to the camera and twodimensional fringe pattern analysis. With SNR lower than 25 dB, subpixel resolution better than 0.3 nm has been accomplished. At the same time high dynamic range on the level of 106 dB has been kept. The author gratefully acknowledges the funding provided by The Ministry of Education, Science and Technological Development of the Republic of Serbia, under projects “Development of methods, sensors and systems for monitoring quality of water, air and soil,” No. III43008, and “Optoelectronic nanodimension systems—route toward applications,” No. III45003. References 1. 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, 6003–6010 (1992). 2. R. Dändliker, E. Zimmermann, and G. Frosio, “Electronically scanned white-light interferometry: a novel noise-resistant signal processing,” Opt. Lett. 17, 679–681 (1992). 3. É. Pinet, E. Cibula, and D. Ðonlagić, “Ultra-miniature allglass Fabry–Pérot pressure sensor manufactured at the tip of a multimode optical fiber,” Proc. SPIE 6770, 67700U (2007). 4. R. V. Neste, C. Belleville, D. Pronovost, and A. Proulx, “System and method for measuring an optical path difference in a sensing interferometer,” U.S. patent 6,842,254 B2 (11 January, 2005). 5. J. Jiang, S. Wang, T. Liu, K. Liu, J. Yin, X. Meng, Y. Zhang, S. Wang, Z. Qin, F. Wu, and D. Li, “A polarized low-coherence interferometry demodulation algorithm by recovering the absolute phase of a selected monochromatic frequency,” Opt. Express 20, 18117–18126 (2012). 6. G. Duplain, “Low-coherence interferometry optical sensor using a single wedge polarization readout interferometer,” U.S. patent 7,259,862 B2 (21 August, 2007). 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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7. http://www.opsens.com/pdf/products/PicoSens%20Rev%201.5 .pdf. 8. C. Ma and A. Wang, “Signal processing of white-light interferometric low-finesse fiber-optic Fabry–Perot sensors,” Appl. Opt. 52, 127–138 (2013). 9. R.-J. Recknagel and G. Notni, “Analysis of white light interferograms using wavelet methods,” Opt. Commun. 148, 122–128 (1998). 10. S. Chen, A. W. Palmer, K. T. V. Grattan, B. T. Meggitt, and S. Martin, “Study of electronically scanned optical-fibre
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white-light Fizeau interferometer,” Electron. Lett. 27, 1032–1034 (1991). 11. R. H. Marshall, Y. N. Ning, X. Jiang, A. W. Palmer, B. T. Meggitt, and K. T. V. Grattan, “A novel electronically scanned white-light interferometer using a Mach– Zehnder approach,” J. Lightwave Technol. 14, 397–402 (1996). 12. J. Park and X. Li, “Theoretical and numerical analysis of superluminescent diodes,” J. Lightwave Technol. 24, 2473– 2480 (2006).
A novel high-resolution wide-dynamic range electronically scanned white-light interferometry-based interrogation technique is presented. By using off-the-shelf optical components, this technique is capable of reaching a subnanometer resolution. The technique relies on a simple optical setup in which the wedge and camera axes are mutually inclined for a very small angle in the horizontal plane and two-dimensional fringe pattern analysis. Resolution below 0.3 nm and dynamic range of 106 dB have been achieved with a signal-to-noise ratio lower than 25 dB. © 2014 Optical Society of America OCIS codes: (060.2370) Fiber optics sensors; (120.3180) Interferometry. http://dx.doi.org/10.1364/AO.53.003341
1. Introduction
Electronically scanned white-light interferometry (WLI) represents a compact, rigid, stable, and fast interrogation technique since there are no mechanical moving parts present in the sensing systems employing this technique. However, this technique suffers from low resolution due to a low signal-tonoise ratio (SNR) and low sampling rate. Therefore, special attention should be paid to the signal processing in order to obtain subpixel resolution. Nevertheless, it is still difficult to reach subnanometer resolution with typical SNR values ranging from 20 to 40 dB [1,2]. A typical application of the electronically scanned WLI is in the signal conditioning of low-finesse fiber-optic Fabry–Perot interferometric sensors [3–6]. A resolution of 0.9 nm in the sensing interferometer path length difference measurement range of 30 μm has been achieved [6,7], thus providing a sensor dynamic range of 90.5 dB. However, in order to obtain this subnanometer resolution a birefringent wedge together with two linear polarizers and focusing optics has been employed. In this paper a novel high-resolution wide-dynamic range 1559-128X/14/153341-06$15.00/0 © 2014 Optical Society of America
electronically scanned WLI-based interrogation technique has been presented that is capable of reaching subnanometer resolution with a SNR lower than 25 dB. The resolution of 0.28 nm, accomplished by the simple setup only consisting of a wedge and camera, is pretty close to the theoretical minimum of 0.17 nm that is obtained for the same SNR [8]. Moreover, this simple configuration is capable of sensing interferometer path length difference measurement in the dynamic range of 106 dB. 2. Theory of Operation
A typical electronically scanned WLI-based sensing system is composed of a collimation lens, a wedge (Fizeau interferometer), and a linear photodetector array. A relatively low number of pixels of the linear array limits the sensor resolution [9]. There are two ways to increase the sensor resolution. The first is to decrease the wedge angle. However, in this way the sensor dynamic range is also decreased. The second way implies the use of linear arrays with a larger number of pixels. This further implies the use of long, custom-made, and expensive linear arrays and wedges that, on the other hand, due to thermal drift and ambient vibration, significantly deteriorate the sensor performances such as stability and accuracy. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
3341
In comparison with a linear array a typical camera has an approximately three orders of magnitude larger number of pixels. Although the camera has a large number of pixels the main problem of involving cameras into the WLI measurement chain is their two-dimensionality. This problem can be overcome if the wedge and camera axes are mutually inclined in a horizontal plane for a small angle. Such a wedge and camera configuration gives the following wedge thickness tx; y above the camera: tx; y t0 x cos φ tan θ y sin φ tan θ ≈ t0 θx φθy;
(1)
where x and y are the coordinates in the camera plain, t0 is the wedge thickness at the coordinate system origin, φφ ≪ 1 is the angle between the wedge and camera axes, and θθ ≪ 1 is the wedge angle. Based on the camera captured two-dimensional lowcoherence interferogram (LCI), a novel algorithm has been developed aimed at finding the central fringe maximum (CFM) position with high resolution. The proposed algorithm consists of two steps. The first step is based on the camera row analysis. Therefore, it is necessary to extract a LCI from a single row, e.g., from a row with the maximal fringe visibility. The wedge thickness along this row positioned at y y0 is given by tx; y0 ≈ t0 φθy0 θx:
(2)
By using one among many algorithms for analyzing LCI [9] one can obtain an estimation of the CFM position. Due to the low SNR and low sampling rate, this estimation is rather poor. Nevertheless, the efficient algorithm will identify at least the position close to the central fringe. When the position, i.e., the pixel in the analyzed row that is in close proximity to the CFM, has been identified, we switch on the second step. In the second step of the algorithm, the camera column that includes this pixel is to be analyzed. Along this column the wedge thickness is given by txCF ; y ≈ t0 θxCF φθy;
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tCFM ≈ t0 θxCF φθyCFM :
(4)
This wedge thickness directly corresponds to the optical path difference (OPD) of the WLI sensing interferometer. Typically, the WLI sensing interferometer OPD is twice as much as the corresponding wedge optical thickness. According to Eq. (4) the single pixel resolution in the measuring WLI sensing interferometer OPD is given by δL ≈ 2npφθ, where n is the wedge index of refraction and p is the distance between the centers of the adjacent camera pixels. In order to get better insight into all relevant geometrical parameters of the wedge and camera setup, in Fig. 1 we have presented the top view of the setup, together with both side views of the setup cross sections along both relevant axes. In Fig. 1, we have also presented the active zone of the camera that represents the rectangle with maximal dimensions on the camera surfaces that is covered by the wedge. Only the images captured by this active zone will be taken into consideration for further signal processing. 3. Experimental Setup
In order to test the capabilities of the proposed WLI interrogation technique a simple low-coherence fiber-optic absolute position sensor has been built. The layout of the experimental setup, based on a single-mode 2 × 2 fiber-optic coupler (FOC), is presented in Fig. 2. The sensing interferometer is a lowfinesse Fabry–Perot interferometer that is formed between the end of the sensing fiber (SF), thus forming the reference mirror (RM) and sensor mirror (SM). To avoid parasitic interference, the distance D between the RM and SM, i.e., the cavity length, must be greater than one half of the used white-light source (WLS) coherence length LC D > LC ∕2. As the WLS a pigtailed superluminescent diode (SLD) is used, which is driven by the commercial current and temperature controller (CTC). SLD emits infrared light with a central wavelength of 842 nm and
(3)
where xCF represents the position of the pixel in close proximity to the central fringe that has been identified in the previous step. If we compare Eqs. (2) and (3) we can notice that wedge thickness changes very slowly along the column providing us an equivalent wedge with a much smaller angle than in the first step. Therefore, a much higher pixel resolution in the CFM positioning can be achieved. In order to reach maximal resolution, angle φ should be chosen as small as possible but still large enough so that each column covers just a couple of fringes. After finding the pixel position yCFM along the observed column that is closest to the CFM that is further closest to the position y y0 of the pixel identified in the 3342
first algorithm step, the wedge thickness tCFM above this particular pixel is given by
Fig. 1. Top view of the setup, together with both side views of the setup cross sections along both relevant axes.
Fig. 2. Layout of the experimental setup.
with a full-width at half-maximum spectral bandwidth of 22 nm. The total optical power launched by the SLD in a single-mode fiber is approximately 1.4 mW. The inactive FOC arm is immersed into the index matching gel (IMG) in order to maximize the interferogram fringe visibility. Finally, the backreflected light is used for wedge (W) illumination. The LCI is captured by the camera (C) and transferred to the personnel computer (PC) for further processing. 4. Results and Discussion
The camera captured images for three distances between RM and SM: (1) D 362 μm, (2) D 370 μm, and (3) D 392 μm are presented in Fig. 3. These distances are within the range in which the highest fringe visibilities of the sensing interferometer have been observed. These two-dimensional LCIs are obtained for φ ≈ 1.84° and θ 0.5°. In order to provide the highest possible level of the interferometric signal, both sides of the wedge are partially reflective with 30% of reflectivity [10]. In Fig. 3, one can clearly notice the wedge edge in the bottom part of each image. Since there are not any coupling optics the wedge illuminating FOC fiber is positioned at a relatively large distance from the wedge surface in order to provide roughly uniform illumination of the camera surface. Moreover, such an arrangement also provides a relatively high spatial coherence of the light beam impinging onto the camera. According to the first step of the proposed algorithm a single row of the captured image should be extracted for further processing. To be able to extract the pixel position/number closest to the CFM the raw data from a single row that are depicted in
Fig. 3. Camera captured two-dimensional LCIs.
Fig. 4(a) should be further processed. The typical signal processing chain of the electronically scanned WLI signal consists of bandpass filtration, normalization with low-pass filtered signal component [11], and envelope detection and fitting. All these signal processing steps are depicted in Figs. 4(b)–4(d), respectively, where lines 1 (circles), 2 (squares), and 3 (triangles) mark the corresponding signals for the following RM–SM distances: D 362 μm,
Fig. 4. Signal processing chain for three RM–SM distances. 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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D 370 μm, and D 392 μm, respectively. The normalization step has been performed in order to suppress the influence of the nonuniform irradiance distribution over the camera surface. The typical SLD has Gaussian spectral distribution [12]. Therefore, the Gaussian function has been used for envelope fitting in the last step. Based on the fitting parameters, where the goodness of fit parameter R2 was greater than 0.98 in all three cases, it is possible to estimate the pixel position/number closest to the CFM position. According to Eq. (3), the corresponding wedge index of refraction n 1.51 and distance between the centers of the adjacent camera pixels p ≈ 2.2 μm, and taking into consideration that WLI sensing interferometer cavity length is equal to the corresponding wedge optical thickness, the single pixel resolution of CFM position estimation in the first algorithm step is equal to ΔD ≈ npθ ≈ 29 nm. This relatively low resolution is still sufficient for identifying the central fringe. After identifying the corresponding pixel number in the first algorithm step we switch on the second step. In order to eliminate the influence of the nonuniform irradiance distribution, bandpass filtration and normalization have been performed over each image row. The rows in the bottom part of the images depicted in Fig. 3, which contain pixels in close proximity to the wedge edge, are omitted from further processing. When filtration and normalization have been performed the column signal that contains the aforementioned pixel should be of the cosine shape. The exact position of the CFM is obtained according to the above presented algorithm in the introductory part of this paper. Based on the values of the experimental setup parameters the estimated overall single pixel resolution in CFM position measurement is δD δL∕2 ≈ 0.93 nm. The overall performances of the proposed algorithm are estimated based on a simple experiment. In the presented experimental setup, where SM was positioned at a distance of approximately 370 μm from RM, SM was slightly shifted with the help of a calibrated high-precision piezo actuator. Both captured images, obtained for the initial and shifted SM positions, were processed according to the presented algorithm. The results, when the SM shift was 5 nm, are presented in Fig. 5. As the result of the first step of the algorithm the Gaussian fitting parameters are obtained, where envelope data and Gaussian fits are presented in Figs. 5(a) and 5(b) for 1 (circle) initial and 2 (square) shifted SM position. Figure 5(b) represents the enlarged view of Fig. 5(a) in close proximity to the maximum value of the fitted function. According to the fitting parameters CFM is positioned with a standard deviation of approximately 1.7 pixels, i.e., with a resolution of approximately 48 nm, where the SNR of the row signal was 24.7 dB. This is still sufficient for central fringe identification, but insufficient to detect the shift of 5 nm. Moreover, the measured SM shift is in the opposite direction from the real one. However, after 3344
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Fig. 5. Measurement process for the SM small shift estimation.
rounding both of the corresponding fitting parameters refer to the same pixel. After identifying the pixel in the inspected row closest to the CFM we switch on the second algorithm step. The corresponding column signals are depicted in Fig. 5(c). One can notice that both signals are overlapping due to the very small SM shift. Therefore, an enlarged view of the Fig. 5(c) central part is presented in Fig. 5(d) in close proximity to the CFM, where lines 1 and 2 represent the initial and shifted SM positions, respectively, and lines 3 and 4 represent their corresponding fitting curves of the cosine shape, where the goodness of fit parameter R2 was greater than 0.987 in both cases. Instead of CFM the central fringe minimum value has been presented in Fig. 5(d). The reason for this lies in the fact that RM and SM have different phase shifts, 0 and π, respectively, after the light beam reflection. From Fig. 5(d) it is evident that the column signal shifts toward the first column pixel. This can be confusing if compared to the row shift obtained in the first algorithm step. However, this is expected as the wedge thickness decreases along the inspected column from the top to the bottom and the first pixel is located at the top of the captured image. Finally, according to the fitted parameters the measured shift is 5.5 nm and its standard deviation is 0.28 nm (0.31 pixels), which at the same time represents the overall sensing system subpixel resolution. In order to estimate the dynamic range of such a built sensing system it will be assumed that the maximally measured cavity length range corresponds to the maximal wedge optical thickness difference along the camera surface. So, for the maximal RM–SM distance range DM we have DM ≈ npN X θ ≈ 59 μm, where N X is the row number of pixels, for which for the used camera we have N X 2048. The dynamic range is defined as the
ratio of the measurement range and resolution. Since SNR changes only for 1.1 dB when the RM–SM distance changes in the range from 362 to 392 μm the resolution of 0.28 nm, which was estimated for the distance of 370 μm, will be taken as the relevant one for the whole measurement range. In this respect the dynamic range is approximately 106 dB. This relatively high dynamic range can be increased if the angle between the wedge and camera decreases. However, there is a limit in decreasing of this angle. In order to cover the entire measurement range with the highest resolution, each column must cover at least the range of a single row pixel; i.e., N Y δD ≥ ΔD must be fulfilled, where N Y is the column number of pixels. In our case we have N Y 1536, where φ ≥ 1∕N Y ≈ 0.037° is valid. Consequently, the maximally theoretically achievable dynamic range of such a composed sensing system is given by 20 log10 DM ∕δD ≈ 20 log10 N X N Y ≈ 130 dB, where the single column pixel resolution has been considered as the relevant resolution of the complete measurement setup. If compared with the single pixel dynamic range of the proposed sensing system of 96 dB, where the achieved single pixel resolution of δD ≈ 0.93 nm has been taken into account, one can notice that there is a space in the sensing system performance improvement. In practice this is difficult to achieve as the angle between the wedge and the camera must be very small, thus reaching higher noise levels and consequently lower resolution. This can be easily seen in Fig. 6(a), where we have presented the power spectral density of the analyzed raw column signal (arrow marks the corresponding carrier signal). If the angle between the wedge and the camera decreases the carrier signal will move toward the lower frequencies where we have higher noise levels due to the flicker noise. Figure 6(b) also presents the power spectral density of the analyzed raw row signal. One can clearly notice the carrier signal marked with an arrow. If the angle between the wedge and the camera is kept very small (φ ≪ 1) there will be no significant change in the carrier signal position. The carrier signal position will be changed only if we chose the wedge with a different angle. Nevertheless, one can notice that the wedge angle has been chosen appropriately as
Fig. 6. Power spectral densities (PSD) of the raw (a) column and (b) row signal.
the carrier signal is positioned away from the high level of flicker noise. In the end, the capability of the first algorithm step to identify the pixel in close proximity to the CFM has been tested. Similarly to the previous test the SM was positioned at a distance of approximately 370 μm from the RM. Since the 568th row of the captured images has been identified as the one with the highest fringe visibility the first algorithm step has been performed over this row. As a result of the first step the 923rd pixel is identified to be the closest one to the CFM. After switching to the second step and analysis of the 923rd column, the 522nd pixel in the inspected column has been identified as the one closest to the CFM. Therefore, an error of 46 column pixels, i.e., an error of 43 nm, occurred in the first algorithm step. This measured error is very close to the previously estimated resolution of the first algorithm step of approximately 48 nm. 5. Conclusion
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