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Wavelength calibration of dispersive near-infrared spectrometer using relative k-space distribution with low coherence interferometer
Optics Communications 367 (2016) 186–191
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelength calibration of dispersive near-infrared spectrometer using relative k-space distribution with low coherence interferometer Ji-hyun Kim, Jae-Ho Han, Jichai Jeong n Department of Brain and Cognitive Engineering, Korea University, 145 Anam-Ro, Seongbuk-ku, Seoul 02841, South Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2015 Received in revised form 13 January 2016 Accepted 17 January 2016 Available online 4 February 2016
The commonly employed calibration methods for laboratory-made spectrometers have several disadvantages, including poor calibration when the number of characteristic spectral peaks is low. Therefore, we present a wavelength calibration method using relative k-space distribution with low coherence interferometer. The proposed method utilizes an interferogram with a perfect sinusoidal pattern in kspace for calibration. Zero-crossing detection extracts the k-space distribution of a spectrometer from the interferogram in the wavelength domain, and a calibration lamp provides information about absolute wavenumbers. To assign wavenumbers, wavelength-to-k-space conversion is required for the characteristic spectrum of the calibration lamp with the extracted k-space distribution. Then, the wavelength calibration is completed by inverse conversion of the k-space into wavelength domain. The calibration performance of the proposed method was demonstrated with two experimental conditions of four and eight characteristic spectral peaks. The proposed method elicited reliable calibration results in both cases, whereas the conventional method of third-order polynomial curve fitting failed to determine wavelengths in the case of four characteristic peaks. Moreover, for optical coherence tomography imaging, the proposed method could improve axial resolution due to higher suppression of sidelobes in point spread function than the conventional method. We believe that our findings can improve not only wavelength calibration accuracy but also resolution for optical coherence tomography. & 2016 Elsevier B.V. All rights reserved.
Keywords: Infrared spectroscopy Optical imaging Biomedical imaging Optical coherence tomography
1. Introduction For spectral domain low coherence interferometry (LCI), dispersive spectrometers have become a critical component for extracting depth information without mechanical scanning. The wavelength profile of the spectrometer in LCI is necessary for converting the spectra from wavelength to a linearly distributed kspace to minimize depth-dependent broadening of the point spread function (PSF) [1–4]. In the case of commercial spectrometers, manufacturers provide wavelength profiles of their products calibrated by tunable lasers and calibration sources. However, laboratory-made spectrometers require manual calibration by the users. Because tunable lasers are relatively expensive, calibration required, and the tuning range presents a limitation [5– 9], the most common tools employed for calibrating dispersive spectrometers are calibration lamps that emit characteristic spectra within specific wavelength ranges. Gratings in dispersive spectrometers are designed to linearly disperse wavelength. However, the wavelength profiles do not n
Corresponding author. E-mail address: [email protected] (J. Jeong).
http://dx.doi.org/10.1016/j.optcom.2016.01.046 0030-4018/& 2016 Elsevier B.V. All rights reserved.
perfectly match the linear dispersion function because of ruling error, focal plane mismatch, optical distortion, and aberration [10– 14]. Therefore, calibrating a spectrometer requires as many characteristic spectral peaks as possible to determine an accurate wavelength profile. Calibration methods based on high-order polynomial fitting generally show acceptable performance, but if there are a limited number of spectral peaks, such methods will completely fail the wavelength calibration. Interferometric wavelength calibration methods for spectrometers in optical coherence tomography (OCT) have been reported. For example, an autocalibration method [15] was proposed for polarization-sensitive OCT. The method was based on the interferometric method and employed a slide glass to generate an interference pattern. However, this method caused a low signal-to-noise ratio and dynamic range loss. Another example of interferometric calibration methods is an automatic spectral calibration method [16] introduced for OCT with mechanical modulation of sample positions. Even though these two methods appropriately compensated for dispersion from nonequal k-space samples, the wavelength profile could not be determined. In addition, an interferometric calibration method [17] was suggested that used a calibration lamp with a microstage for wavelength calibration, but it required iterative experiments to accurately determine the wavelength profile.
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
187
Fig. 1. Schematic of fiber-based low coherence interferometry.
Fig. 2. Detected characteristic spectrum of an argon lamp by the spectrometer.
Against this background, we propose a new calibration method to reduce calibration errors. This method utilizes an interference pattern with a perfect sinusoidal function in an equally distributed k-space. To extract k-space distribution in the wavelength domain, we performed zero-crossing detection and converted the interferogram in wavelength domain into a linearly distributed k-space. We call the extracted k-space distribution as relative k-space distribution because it does not include absolute wavenumbers. Without wavelength calibration, conversion with relative k-space distribution allowed minimization of depth-dependent dispersion in an OCT image. For wavelength calibration of a spectrometer, we used the characteristic spectral peaks of a calibration lamp. After converting the characteristic spectrum into an equally distributed k-space using the relative kspace distribution, we could assign absolute wavenumbers. Then, the wavelength calibration could be completed by inverse conversion of the k-space into the wavelength domain by using the relative k-space distribution again. We demonstrate the performance of the proposed method against a conventional method by comparison with ground truth (the calibration profile from the manufacturer), and show that the method exhibits reliable calibration performance.
2. Instrumentation and measurement 2.1. Wavelength calibration of dispersive spectrometer using relative k-space distribution with LCI LCI consists of a broadband light source, a detector, and the
Fig. 3. (a) Recorded spectrum in the first region without sample mirror. (b) Interference in the original spectrum after placing the sample mirror with arbitrary optical delay. (c) Eliminating DC terms for zero-crossing detection.
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J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
Fig. 4. Calibration process with the proposed method. (a) A DC-rejected interferogram using reference subtraction and high-pass filtering. (b) Relative k-space distribution profile determined by zero-crossing detection. (c) Interpolated interferogram in linearly distributed k-space. (d) Detected characteristic spectral peaks of the Ar lamp. (e) Interpolated spectrum of the Ar lamp using the relative k-space distribution. (f) Calibrated wavelength profile of the spectrometer after interpolation with wavenumberto-wavelength conversion.
Michelson interferometer. Fig. 1 shows a fiber-based LCI that includes a light source, fiber coupler, reference mirror, sample mirror, and spectrometer for detecting the interferogram. The illuminated light from the low coherence source is split by the 50/50 fiber coupler and directed toward both reference and sample mirrors. The position of the sample mirror can be adjusted to control the optical delay of the back-reflected fields from the reference and the sample arms. The coupler combines the back-reflected fields and generates interference patterns, and the intensity detected by the spectrometer can be defined as
{
I (k ) = ρS (k ) Is + Ir +
Is Ir (e j2kd + e−j2kd )
}
(1)
where Ir and Is are the light intensity from the reference and the sample mirrors, respectively, ρ is the responsivity of the detector, S (k) is the light source spectrum, and d is the path difference between the sample and the reference arms. Using Euler's formula, (1) can be rewritten as
{
I (k ) = 2ρS (k ) Is + Ir +
Is Ir cos (2kd)
}
(2)
The interferogram includes a perfect sinusoidal function in terms of k. By applying an inverse Fourier transform to (2) after rejecting DC signals, the optical path difference can be extracted as well. By using this characteristic, we can inversely determine the k-space distribution of a spectrometer. Initially, to determine k, the broadband light source should cover the full bandwidth of the spectrometer for calibration. Otherwise, the calibration accuracy will be significantly low because wavenumbers outside the calibration range are estimated. After an interference pattern is obtained, we can obtain the relative k-space distribution by zero-crossing detection [15]. The relative k-space distribution represents a linearly distributed kspace profile of the spectrometer in the wavelength domain. Therefore, the rate of change in the k-values is determined by remapping the relative k-space distribution to the linearly distributed k-space. To map the absolute wavenumbers to the relative
k-space distribution, at least two absolute wavenumbers are required. The wavenumbers may be acquired from calibration lamps and lasers. The k-values provide information about the step size of k (dk) as well as the absolute wavenumbers on specific camera pixels of the spectrometer. Finally, the spectrometer can be calibrated with the k-values, dk, and relative k-space distribution. 2.2. Experimental system configuration We implemented a fiber-based LCI using a multiplexed superluminescent light emitting diode (SLD) source with four separate SLDs [18]. The light source had a full width half maximum (FWHM) bandwidth of 220 nm centered at 1375 nm. An InGaAs line camera attached to a spectrometer (BaySpec, USA) supported a maximum line rate of 92 kHz with 1024 pixels, and the nominal focal plane array sensitivity was set to 450 e-/counts. The spectrometer covered a wavelength range of 1254–1487 nm. It was calibrated by the manufacturer, and we used the calibration profile as the ground truth for experimental comparison. Two silver-coated mirrors were placed at both output arms. Both paths had arbitrary optical delays to generate a sinusoidal fringe pattern. To extract the absolute wavenumbers, we used 10 characteristic spectral peaks of an argon (Ar) lamp at 1280.274 nm, 1295.666 nm, 1327.264 nm, 1331.321 nm, 1336.711 nm, 1350.419 nm, 1362.266 nm, 1367.855 nm, 1371.858 nm, and 1409.364 nm. Fig. 2 shows the characteristic spectrum of the Ar lamp detected by the spectrometer.
3. Results and discussion To demonstrate the performance of the proposed calibration method, we introduced two experimental conditions: a large and a small number of characteristic spectral peaks in the region of interest for calibration. To ensure that the experimental conditions in the experiments were similar, we used a single spectrometer and divided the detection region of the spectrometer into two, each
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
189
Fig. 5. (a) Characteristic spectral peaks for calibration and (b) wavelength calibration result from the first region (black dots represent characteristic spectral peaks).
Fig. 6. (a) Characteristic spectral peaks for calibration and (b) wavelength calibration result from the second region (black dot represent characteristic spectral peaks).
with 512 pixels. The pixel range of 80–591 had the maximum number of characteristic spectral peaks, but we did not select that range because 100 pixels from both the start and the end of the spectrometer (pixel range of 1–100 and 925–1024) included insufficient fringe intensity for zero-crossing detection because of the low level of the power of SLDs. Therefore, to obtain a sufficient number of characteristic spectral peaks with sufficient fringe intensity, we chose the pixel ranges for experiments to be 110–621 and 410–921. The first region included eight characteristic spectral peaks of 1295.666 nm, 1327.264 nm, 1331.321 nm, 1336.711 nm, 1350.419 nm, 1362.266 nm, 1367.855 nm, and 1371.858 nm at pixel indices of 174, 308, 326, 349, 408, 459, 483, and 501, respectively. The second region had only four characteristic spectral peaks of 1362.266 nm, 1367.855 nm, 1371.858 nm, and 1409.364 nm at pixel indices of 459, 483, 501, and 666, respectively. Moreover, the distribution of the characteristic spectral peaks in the second region was concentrated in the first half of the spectral range, whereas the first region had evenly distributed characteristic spectral peaks. For third-order polynomial curve fitting, if there were more than three characteristic spectral peaks, an even distribution of the peaks was more important than the number of the peaks.
Fig. 3(a) shows the original spectrum of the light source in the first region. When the sample mirror was placed in the sample arm with an arbitrary optical delay, a fringe pattern occurred in the spectrum as shown in Fig. 3(b). Prior to zero-crossing detection, the DC terms should be eliminated. We applied a reference subtraction method [19] and high-pass filtering for DC rejection (Fig. 3(c)) to reduce the detection error of relative k-space distribution. Once the DC-rejected spectrum was obtained as shown in Fig. 4(a), zero-crossing detection extracted the relative k-space distribution on the pixel indices (Fig. 4(b)). Spectral domain LCI requires interpolation to linearize the interferogram in k-space using the relative k-space distribution (Fig. 4(c)) to reduce depthdependent dispersion. However, we could not finalize the wavelength calibration at this point because the relative k-space distribution did not provide the absolute wavenumbers on camera pixels. Therefore, we introduced a calibration lamp to determine wavenumbers from characteristic spectral peaks. Fig. 4(d) shows the characteristic spectral peaks of the Ar lamp. By using interpolation with the relative k-space distribution, the spectrum (Fig. 4 (d)) was assigned to a linearized k-space as shown in Fig. 4(e).
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Fig. 7. Normalized point spread functions from (a) the first region and (b) the second region.
From the linearized spectrum (Fig. 4(e)), we could estimate dk with
dk =
⎡ N−1 M 2 kn − km ⎢∑ ∑ N (N − 1) ⎢⎣ n = 1 m = n + 1 pn − pm
⎤ ⎥ ⎥⎦
(3)
where N and M are the numbers of characteristic spectral peaks, k is the wavenumber on a pixel, and p is a pixel number with running integers of n and m. The wavenumber spacing between camera pixels in the spectrometer was estimated using this equation. In addition, the exact wavenumbers on pixel indices were also obtained, which allowed calibration of the spectrometer in k-space as well as in the wavelength domain with k-to-λ conversion (Fig. 4(f)). To optimize the dk calculation, we eliminated outliers. Even though the calculation accuracy improved, the improvement was limited to below 1%. This was because the relative k-space distribution was already smoothed by curve fitting that prevented the occurrence of significant outliers during the calculation. Moreover, the application of median and grouping spectral peaks was attempted for optimization; however, this sometimes produced
erroneous results because of an insufficient number of dk samples for calculation. To evaluate how the calibration method performs in comparison with conventional methods, we calibrated the spectrometer using the proposed method and a conventional method. The conventional method fitted the characteristic spectral peaks of the calibration lamp with a third-order polynomial curve for wavelength calibration. The proposed method utilized not only an interference pattern but also the characteristic spectral peaks, as described in the previous section. We used the factory-calibrated wavelength profile as the ground truth. Fig. 5 shows the results for the first region. The root mean square error (RMSE) between the proposed method and the ground truth was 0.103 nm, whereas it was 0.113 nm in the case of the conventional method. The dk estimated by the proposed method and the conventional method were 815.819 m 1 and 816.386 m 1, respectively, with the ground truth dk being 815.191 m 1. Both the methods showed reliable and comparable calibration performance because a sufficient number of characteristic spectral peaks were available for calibration. The second region had only four characteristic spectral peaks as shown in Fig. 6(a). In this case, the conventional method failed to calibrate the wavelengths of the spectrometer in the pixel range of 300–500 because information about the wavelengths in that range was insufficient. The proposed method successively followed the ground truth profile, whereas the conventional method caused significant errors after 1410 nm (Fig. 6(b)). The measured RMSEs were 0.425 nm and 21.155 nm for the proposed method and conventional method, respectively. The dk estimated by the proposed method and conventional method were 713.117 m 1 and 1089.823 m 1, respectively, with the ground truth dk being 711.286 m 1. The RMSE and estimation error of dk of the proposed method were slightly higher than those of the first experiment because there were a limited number of known wavenumbers to precisely estimate the wavelength and dk. Still, the proposed method showed reliable calibration performance and closely matched the ground truth. The proposed method produced slightly larger dk values than the values obtained from the ground truth in the two experiments. We calculated the ground truth profile using third-order polynomial curve fitting with information on nine spectral peaks provided from the manufacturer. This calibration procedure was exactly the same as in the conventional method, so the ground truth probably included a calibration error within acceptable tolerances. Indeed, in the range of 110–621, the RMSEs between the calibrated wavelengths and the characteristic spectra were 0.143 nm and 0.081 nm for the ground truth and the proposed method, respectively. This result showed that the proposed method could provide more accurate calibration performance than the conventional method. Furthermore, the calibration of a spectrometer affects the PSF in spectral domain OCT because wavelength needs to be converted to linearly distributed k-space to prevent depth-dependent dispersion [20,21]. To demonstrate the calibration effect on PSF, we obtained PSFs (Fig. 7) by inverse Fourier transform of the calibration profiles from the first and second regions. The proposed and conventional methods had similar RMSEs for the first region, but the proposed method showed a narrower PSF than the conventional method (Fig. 7(a)); the PSFs from both methods closely followed the ground truth PSF. The measured FWHM resolutions were 7.52 μm and 12.72 μm for the proposed and conventional methods, respectively, and the proposed method showed better performance by 59% than that from the conventional method. For the second region, the PSF from the conventional method broadened significantly whereas the proposed method showed almost the same PSF as the ground truth PSF (Fig. 7(b)). These results
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
show that the proposed method effectively determined an accurate calibration profile for OCT imaging as well as the wavelengths.
4. Conclusion In this paper, we proposed and demonstrated an interferometric calibration method for dispersive spectrometers. The proposed method utilizes a calibration lamp and LCI. In the calibration system, an interferogram provides information about the relative k-space distribution by zero-crossing detection that represents how wavenumbers change in the wavelength domain. This relative k-space distribution could be directly used for OCT imaging to reduce depth-dependent dispersion, but wavelength calibration was not possible because the distribution did not provide any information about wavenumbers or dk of the spectrometer. Therefore, we measured the wavenumbers and dk from a calibration lamp, and at least two exact wavenumbers or wavelengths were necessary for calibration. To demonstrate the performance of the proposed method, we performed two experiments with four and eight characteristic spectral peaks and compared the performance with that of a conventional method. With eight characteristic peaks, the proposed method and conventional method successfully performed wavelength calibration, but the resolution of OCT was better with the proposed method. In the case of four characteristic spectral peaks, the conventional method failed the calibration but the proposed method successfully determined the wavelengths. We believe that our findings can improve not only wavelength calibration accuracy but also resolution for OCT.
Acknowledgment This research was supported in part by a Korea University Grant, the Brain Korea 21 PLUS Program through the National Research Foundation of Korea funded by the Ministry of Education, and by the Basic Science Research Program through the National Research Foundation of Korea (2013R1A1A2062448).
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[3] N. Zhang, T.Y. Chen, C.M. Wang, J. Zhang, T.C. Huo, J.G. Zheng, P. Xue, Spectraldomain optical coherence tomography with a Fresnel spectrometer, Opt. Lett. 37 (2012) 1307–1309. [4] N. Zhang, T.C. Huo, C.M. Wang, T.Y. Chen, J.G. Zheng, P. Xue, Compressed sensing with linear-in-wavenumber sampling in spectral-domain optical coherence tomography, Opt. Lett. 37 (2012) 3075–3077. [5] M.A. Gil, J.M. Simon, A.N. Fantino, Czerny–Turner spectrograph with a wide spectral range, Appl. Opt. 27 (1988) 4069–4072. [6] D.R. Lobb, Theory of concentric designs for grating spectrometers, Appl. Opt. 33 (1994) 2648–2658. [7] B. Shafer, L.R. Megill, L. Droppleman, Optimization of the Czerny–Turner spectrometer, J. Opt. Soc. Am. 54 (1964) 879–886. [8] Q.S. Xue, S.R. Wang, F.Q. Lu, Aberration-corrected Czerny–Turner imaging spectrometer with a wide spectral region, Appl. Opt. 48 (2009) 11–16. [9] K.W. Busch, D. Rabbe, K. Humphrey, M.A. Busch, Design and evaluation of a near-infrared dispersive spectrometer that uses a He–Ne laser for automatic internal wavelength calibration, Appl. Spectrosc. 56 (2002) 346–349. [10] G. Sarlet, G. Morthier, R. Baets, Control of widely tunable SSG-DBR lasers for dense wavelength division multiplexing, J. Lightwave Technol. 18 (2000) 1128–1138. [11] C. Ding, P. Bu, X.Z. Wang, O. Sasaki, A new spectral calibration method for Fourier domain optical coherence tomography, Optik 121 (2010) 965–970. [12] Y.R. Chen, B. Sun, T. Han, Y.F. Kong, C.H. Xu, P. Zhou, X.F. Li, S.Y. Wang, Y. X. Zheng, L.Y. Chen, Densely folded spectral images of a CCD spectrometer working in the full 200–1000 nm wavelength range with high resolution, Opt. Express 13 (2005) 10049–10054. [13] R. Leitgeb, C.K. Hitzenberger, A.F. Fercher, Performance of fourier domain vs. time domain optical coherence tomography, Opt. Express 11 (2003) 889–894. [14] L.A. Coldren, G.A. Fish, Y. Akulova, J.S. Barton, L. Johansson, C.W. Coldren, Tunable semiconductor lasers: a tutorial, J. Lightwave Technol. 22 (2004) 193–202. [15] M. Mujat, B.H. Park, B. Cense, T.C. Chen, J.F. de Boer, Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination, J. Biomed. Opt. 12 (2007) 041205. [16] X.A. Liu, M. Balicki, R.H. Taylor, J.U. Kang, Towards automatic calibration of fourier-domain OCT for robot-assisted vitreoretinal surgery, Opt. Express 18 (2010) 24331–24343. [17] J.H. Kim, J.H. Han, J. Jeong, Accurate wavelength calibration method for spectrometer using low coherence interferometry, J. Lightwave Technol. 33 (2015) 3413–3418. [18] J.H. Kim, J.H. Han, J. Jeong, Graphics processing unit aided highly stable realtime spectral-domain optical coherence tomography at 1375 nm based on dual-coupled-line subtraction, Laser Phys. Lett. 10 (2013) 045604. [19] J.H. Kim, J.H. Han, J. Jeong, Periodic reference subtraction method for efficient background fixed pattern noise removal in Fourier domain optical coherence tomography, Opt. Commun. 285 (2012) 2012–2016. [20] S. Van der Jeught, A. Bradu, A.G. Podoleanu, Real-time resampling in Fourier domain optical coherence tomography using a graphics processing unit, J. Biomed. Opt. 15 (2010) 030511. [21] C. Chong, A. Morosawa, T. Sakai, High-speed wavelength-swept laser source with high-linearity sweep for optical coherence tomography, IEEE J. Sel. Top. Quantum Electron. 14 (2008) 235–242.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Wavelength calibration of dispersive near-infrared spectrometer using relative k-space distribution with low coherence interferometer Ji-hyun Kim, Jae-Ho Han, Jichai Jeong n Department of Brain and Cognitive Engineering, Korea University, 145 Anam-Ro, Seongbuk-ku, Seoul 02841, South Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2015 Received in revised form 13 January 2016 Accepted 17 January 2016 Available online 4 February 2016
The commonly employed calibration methods for laboratory-made spectrometers have several disadvantages, including poor calibration when the number of characteristic spectral peaks is low. Therefore, we present a wavelength calibration method using relative k-space distribution with low coherence interferometer. The proposed method utilizes an interferogram with a perfect sinusoidal pattern in kspace for calibration. Zero-crossing detection extracts the k-space distribution of a spectrometer from the interferogram in the wavelength domain, and a calibration lamp provides information about absolute wavenumbers. To assign wavenumbers, wavelength-to-k-space conversion is required for the characteristic spectrum of the calibration lamp with the extracted k-space distribution. Then, the wavelength calibration is completed by inverse conversion of the k-space into wavelength domain. The calibration performance of the proposed method was demonstrated with two experimental conditions of four and eight characteristic spectral peaks. The proposed method elicited reliable calibration results in both cases, whereas the conventional method of third-order polynomial curve fitting failed to determine wavelengths in the case of four characteristic peaks. Moreover, for optical coherence tomography imaging, the proposed method could improve axial resolution due to higher suppression of sidelobes in point spread function than the conventional method. We believe that our findings can improve not only wavelength calibration accuracy but also resolution for optical coherence tomography. & 2016 Elsevier B.V. All rights reserved.
Keywords: Infrared spectroscopy Optical imaging Biomedical imaging Optical coherence tomography
1. Introduction For spectral domain low coherence interferometry (LCI), dispersive spectrometers have become a critical component for extracting depth information without mechanical scanning. The wavelength profile of the spectrometer in LCI is necessary for converting the spectra from wavelength to a linearly distributed kspace to minimize depth-dependent broadening of the point spread function (PSF) [1–4]. In the case of commercial spectrometers, manufacturers provide wavelength profiles of their products calibrated by tunable lasers and calibration sources. However, laboratory-made spectrometers require manual calibration by the users. Because tunable lasers are relatively expensive, calibration required, and the tuning range presents a limitation [5– 9], the most common tools employed for calibrating dispersive spectrometers are calibration lamps that emit characteristic spectra within specific wavelength ranges. Gratings in dispersive spectrometers are designed to linearly disperse wavelength. However, the wavelength profiles do not n
Corresponding author. E-mail address: [email protected] (J. Jeong).
http://dx.doi.org/10.1016/j.optcom.2016.01.046 0030-4018/& 2016 Elsevier B.V. All rights reserved.
perfectly match the linear dispersion function because of ruling error, focal plane mismatch, optical distortion, and aberration [10– 14]. Therefore, calibrating a spectrometer requires as many characteristic spectral peaks as possible to determine an accurate wavelength profile. Calibration methods based on high-order polynomial fitting generally show acceptable performance, but if there are a limited number of spectral peaks, such methods will completely fail the wavelength calibration. Interferometric wavelength calibration methods for spectrometers in optical coherence tomography (OCT) have been reported. For example, an autocalibration method [15] was proposed for polarization-sensitive OCT. The method was based on the interferometric method and employed a slide glass to generate an interference pattern. However, this method caused a low signal-to-noise ratio and dynamic range loss. Another example of interferometric calibration methods is an automatic spectral calibration method [16] introduced for OCT with mechanical modulation of sample positions. Even though these two methods appropriately compensated for dispersion from nonequal k-space samples, the wavelength profile could not be determined. In addition, an interferometric calibration method [17] was suggested that used a calibration lamp with a microstage for wavelength calibration, but it required iterative experiments to accurately determine the wavelength profile.
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
187
Fig. 1. Schematic of fiber-based low coherence interferometry.
Fig. 2. Detected characteristic spectrum of an argon lamp by the spectrometer.
Against this background, we propose a new calibration method to reduce calibration errors. This method utilizes an interference pattern with a perfect sinusoidal function in an equally distributed k-space. To extract k-space distribution in the wavelength domain, we performed zero-crossing detection and converted the interferogram in wavelength domain into a linearly distributed k-space. We call the extracted k-space distribution as relative k-space distribution because it does not include absolute wavenumbers. Without wavelength calibration, conversion with relative k-space distribution allowed minimization of depth-dependent dispersion in an OCT image. For wavelength calibration of a spectrometer, we used the characteristic spectral peaks of a calibration lamp. After converting the characteristic spectrum into an equally distributed k-space using the relative kspace distribution, we could assign absolute wavenumbers. Then, the wavelength calibration could be completed by inverse conversion of the k-space into the wavelength domain by using the relative k-space distribution again. We demonstrate the performance of the proposed method against a conventional method by comparison with ground truth (the calibration profile from the manufacturer), and show that the method exhibits reliable calibration performance.
2. Instrumentation and measurement 2.1. Wavelength calibration of dispersive spectrometer using relative k-space distribution with LCI LCI consists of a broadband light source, a detector, and the
Fig. 3. (a) Recorded spectrum in the first region without sample mirror. (b) Interference in the original spectrum after placing the sample mirror with arbitrary optical delay. (c) Eliminating DC terms for zero-crossing detection.
188
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
Fig. 4. Calibration process with the proposed method. (a) A DC-rejected interferogram using reference subtraction and high-pass filtering. (b) Relative k-space distribution profile determined by zero-crossing detection. (c) Interpolated interferogram in linearly distributed k-space. (d) Detected characteristic spectral peaks of the Ar lamp. (e) Interpolated spectrum of the Ar lamp using the relative k-space distribution. (f) Calibrated wavelength profile of the spectrometer after interpolation with wavenumberto-wavelength conversion.
Michelson interferometer. Fig. 1 shows a fiber-based LCI that includes a light source, fiber coupler, reference mirror, sample mirror, and spectrometer for detecting the interferogram. The illuminated light from the low coherence source is split by the 50/50 fiber coupler and directed toward both reference and sample mirrors. The position of the sample mirror can be adjusted to control the optical delay of the back-reflected fields from the reference and the sample arms. The coupler combines the back-reflected fields and generates interference patterns, and the intensity detected by the spectrometer can be defined as
{
I (k ) = ρS (k ) Is + Ir +
Is Ir (e j2kd + e−j2kd )
}
(1)
where Ir and Is are the light intensity from the reference and the sample mirrors, respectively, ρ is the responsivity of the detector, S (k) is the light source spectrum, and d is the path difference between the sample and the reference arms. Using Euler's formula, (1) can be rewritten as
{
I (k ) = 2ρS (k ) Is + Ir +
Is Ir cos (2kd)
}
(2)
The interferogram includes a perfect sinusoidal function in terms of k. By applying an inverse Fourier transform to (2) after rejecting DC signals, the optical path difference can be extracted as well. By using this characteristic, we can inversely determine the k-space distribution of a spectrometer. Initially, to determine k, the broadband light source should cover the full bandwidth of the spectrometer for calibration. Otherwise, the calibration accuracy will be significantly low because wavenumbers outside the calibration range are estimated. After an interference pattern is obtained, we can obtain the relative k-space distribution by zero-crossing detection [15]. The relative k-space distribution represents a linearly distributed kspace profile of the spectrometer in the wavelength domain. Therefore, the rate of change in the k-values is determined by remapping the relative k-space distribution to the linearly distributed k-space. To map the absolute wavenumbers to the relative
k-space distribution, at least two absolute wavenumbers are required. The wavenumbers may be acquired from calibration lamps and lasers. The k-values provide information about the step size of k (dk) as well as the absolute wavenumbers on specific camera pixels of the spectrometer. Finally, the spectrometer can be calibrated with the k-values, dk, and relative k-space distribution. 2.2. Experimental system configuration We implemented a fiber-based LCI using a multiplexed superluminescent light emitting diode (SLD) source with four separate SLDs [18]. The light source had a full width half maximum (FWHM) bandwidth of 220 nm centered at 1375 nm. An InGaAs line camera attached to a spectrometer (BaySpec, USA) supported a maximum line rate of 92 kHz with 1024 pixels, and the nominal focal plane array sensitivity was set to 450 e-/counts. The spectrometer covered a wavelength range of 1254–1487 nm. It was calibrated by the manufacturer, and we used the calibration profile as the ground truth for experimental comparison. Two silver-coated mirrors were placed at both output arms. Both paths had arbitrary optical delays to generate a sinusoidal fringe pattern. To extract the absolute wavenumbers, we used 10 characteristic spectral peaks of an argon (Ar) lamp at 1280.274 nm, 1295.666 nm, 1327.264 nm, 1331.321 nm, 1336.711 nm, 1350.419 nm, 1362.266 nm, 1367.855 nm, 1371.858 nm, and 1409.364 nm. Fig. 2 shows the characteristic spectrum of the Ar lamp detected by the spectrometer.
3. Results and discussion To demonstrate the performance of the proposed calibration method, we introduced two experimental conditions: a large and a small number of characteristic spectral peaks in the region of interest for calibration. To ensure that the experimental conditions in the experiments were similar, we used a single spectrometer and divided the detection region of the spectrometer into two, each
J.-h. Kim et al. / Optics Communications 367 (2016) 186–191
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Fig. 5. (a) Characteristic spectral peaks for calibration and (b) wavelength calibration result from the first region (black dots represent characteristic spectral peaks).
Fig. 6. (a) Characteristic spectral peaks for calibration and (b) wavelength calibration result from the second region (black dot represent characteristic spectral peaks).
with 512 pixels. The pixel range of 80–591 had the maximum number of characteristic spectral peaks, but we did not select that range because 100 pixels from both the start and the end of the spectrometer (pixel range of 1–100 and 925–1024) included insufficient fringe intensity for zero-crossing detection because of the low level of the power of SLDs. Therefore, to obtain a sufficient number of characteristic spectral peaks with sufficient fringe intensity, we chose the pixel ranges for experiments to be 110–621 and 410–921. The first region included eight characteristic spectral peaks of 1295.666 nm, 1327.264 nm, 1331.321 nm, 1336.711 nm, 1350.419 nm, 1362.266 nm, 1367.855 nm, and 1371.858 nm at pixel indices of 174, 308, 326, 349, 408, 459, 483, and 501, respectively. The second region had only four characteristic spectral peaks of 1362.266 nm, 1367.855 nm, 1371.858 nm, and 1409.364 nm at pixel indices of 459, 483, 501, and 666, respectively. Moreover, the distribution of the characteristic spectral peaks in the second region was concentrated in the first half of the spectral range, whereas the first region had evenly distributed characteristic spectral peaks. For third-order polynomial curve fitting, if there were more than three characteristic spectral peaks, an even distribution of the peaks was more important than the number of the peaks.
Fig. 3(a) shows the original spectrum of the light source in the first region. When the sample mirror was placed in the sample arm with an arbitrary optical delay, a fringe pattern occurred in the spectrum as shown in Fig. 3(b). Prior to zero-crossing detection, the DC terms should be eliminated. We applied a reference subtraction method [19] and high-pass filtering for DC rejection (Fig. 3(c)) to reduce the detection error of relative k-space distribution. Once the DC-rejected spectrum was obtained as shown in Fig. 4(a), zero-crossing detection extracted the relative k-space distribution on the pixel indices (Fig. 4(b)). Spectral domain LCI requires interpolation to linearize the interferogram in k-space using the relative k-space distribution (Fig. 4(c)) to reduce depthdependent dispersion. However, we could not finalize the wavelength calibration at this point because the relative k-space distribution did not provide the absolute wavenumbers on camera pixels. Therefore, we introduced a calibration lamp to determine wavenumbers from characteristic spectral peaks. Fig. 4(d) shows the characteristic spectral peaks of the Ar lamp. By using interpolation with the relative k-space distribution, the spectrum (Fig. 4 (d)) was assigned to a linearized k-space as shown in Fig. 4(e).
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Fig. 7. Normalized point spread functions from (a) the first region and (b) the second region.
From the linearized spectrum (Fig. 4(e)), we could estimate dk with
dk =
⎡ N−1 M 2 kn − km ⎢∑ ∑ N (N − 1) ⎢⎣ n = 1 m = n + 1 pn − pm
⎤ ⎥ ⎥⎦
(3)
where N and M are the numbers of characteristic spectral peaks, k is the wavenumber on a pixel, and p is a pixel number with running integers of n and m. The wavenumber spacing between camera pixels in the spectrometer was estimated using this equation. In addition, the exact wavenumbers on pixel indices were also obtained, which allowed calibration of the spectrometer in k-space as well as in the wavelength domain with k-to-λ conversion (Fig. 4(f)). To optimize the dk calculation, we eliminated outliers. Even though the calculation accuracy improved, the improvement was limited to below 1%. This was because the relative k-space distribution was already smoothed by curve fitting that prevented the occurrence of significant outliers during the calculation. Moreover, the application of median and grouping spectral peaks was attempted for optimization; however, this sometimes produced
erroneous results because of an insufficient number of dk samples for calculation. To evaluate how the calibration method performs in comparison with conventional methods, we calibrated the spectrometer using the proposed method and a conventional method. The conventional method fitted the characteristic spectral peaks of the calibration lamp with a third-order polynomial curve for wavelength calibration. The proposed method utilized not only an interference pattern but also the characteristic spectral peaks, as described in the previous section. We used the factory-calibrated wavelength profile as the ground truth. Fig. 5 shows the results for the first region. The root mean square error (RMSE) between the proposed method and the ground truth was 0.103 nm, whereas it was 0.113 nm in the case of the conventional method. The dk estimated by the proposed method and the conventional method were 815.819 m 1 and 816.386 m 1, respectively, with the ground truth dk being 815.191 m 1. Both the methods showed reliable and comparable calibration performance because a sufficient number of characteristic spectral peaks were available for calibration. The second region had only four characteristic spectral peaks as shown in Fig. 6(a). In this case, the conventional method failed to calibrate the wavelengths of the spectrometer in the pixel range of 300–500 because information about the wavelengths in that range was insufficient. The proposed method successively followed the ground truth profile, whereas the conventional method caused significant errors after 1410 nm (Fig. 6(b)). The measured RMSEs were 0.425 nm and 21.155 nm for the proposed method and conventional method, respectively. The dk estimated by the proposed method and conventional method were 713.117 m 1 and 1089.823 m 1, respectively, with the ground truth dk being 711.286 m 1. The RMSE and estimation error of dk of the proposed method were slightly higher than those of the first experiment because there were a limited number of known wavenumbers to precisely estimate the wavelength and dk. Still, the proposed method showed reliable calibration performance and closely matched the ground truth. The proposed method produced slightly larger dk values than the values obtained from the ground truth in the two experiments. We calculated the ground truth profile using third-order polynomial curve fitting with information on nine spectral peaks provided from the manufacturer. This calibration procedure was exactly the same as in the conventional method, so the ground truth probably included a calibration error within acceptable tolerances. Indeed, in the range of 110–621, the RMSEs between the calibrated wavelengths and the characteristic spectra were 0.143 nm and 0.081 nm for the ground truth and the proposed method, respectively. This result showed that the proposed method could provide more accurate calibration performance than the conventional method. Furthermore, the calibration of a spectrometer affects the PSF in spectral domain OCT because wavelength needs to be converted to linearly distributed k-space to prevent depth-dependent dispersion [20,21]. To demonstrate the calibration effect on PSF, we obtained PSFs (Fig. 7) by inverse Fourier transform of the calibration profiles from the first and second regions. The proposed and conventional methods had similar RMSEs for the first region, but the proposed method showed a narrower PSF than the conventional method (Fig. 7(a)); the PSFs from both methods closely followed the ground truth PSF. The measured FWHM resolutions were 7.52 μm and 12.72 μm for the proposed and conventional methods, respectively, and the proposed method showed better performance by 59% than that from the conventional method. For the second region, the PSF from the conventional method broadened significantly whereas the proposed method showed almost the same PSF as the ground truth PSF (Fig. 7(b)). These results
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show that the proposed method effectively determined an accurate calibration profile for OCT imaging as well as the wavelengths.
4. Conclusion In this paper, we proposed and demonstrated an interferometric calibration method for dispersive spectrometers. The proposed method utilizes a calibration lamp and LCI. In the calibration system, an interferogram provides information about the relative k-space distribution by zero-crossing detection that represents how wavenumbers change in the wavelength domain. This relative k-space distribution could be directly used for OCT imaging to reduce depth-dependent dispersion, but wavelength calibration was not possible because the distribution did not provide any information about wavenumbers or dk of the spectrometer. Therefore, we measured the wavenumbers and dk from a calibration lamp, and at least two exact wavenumbers or wavelengths were necessary for calibration. To demonstrate the performance of the proposed method, we performed two experiments with four and eight characteristic spectral peaks and compared the performance with that of a conventional method. With eight characteristic peaks, the proposed method and conventional method successfully performed wavelength calibration, but the resolution of OCT was better with the proposed method. In the case of four characteristic spectral peaks, the conventional method failed the calibration but the proposed method successfully determined the wavelengths. We believe that our findings can improve not only wavelength calibration accuracy but also resolution for OCT.
Acknowledgment This research was supported in part by a Korea University Grant, the Brain Korea 21 PLUS Program through the National Research Foundation of Korea funded by the Ministry of Education, and by the Basic Science Research Program through the National Research Foundation of Korea (2013R1A1A2062448).
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