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Abstract
A new method for spectroscopic interferometry using rotating diffraction grating was developed for industrial measurements. Two diffraction gratings increase the spectroscopic resolution, and the effective measuring range can be extended considerably. Instead of calibrating the wavelength, we used the Fabry−Perot Etalon (standard) to calibrate the system and determine the absolute position. The rotation diffraction gratings may also be used as a spectroscopic element over extensive ranges for low-cost and high-speed measurement. Our experiments indicate a length range of approximately 4.00 mm with repeatability of 0.17μm (0.0167%) for the narrow range and 3.84 μm (0.0955%) for the wide range.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spectroscopy is an important measurement technique that has been applied to virtually all fields of science and technology [1–5]. The spectral interferogram includes background component related to the light source spectrum and useful component of interference fringes related to the spectral wavelength difference between the interfering light beams. The information about the optical path difference to be measured is contained in unwrapped fringe phase and it can be used for measuring thickness [6,7] and in optical profilometry [8–10]. Spectrometer characterization, such as wavelength calibration and resolution, influence the operation of the device [11]. Dispersive spectrometers are subject to wavelength dispersion errors due to aberration, ruling errors, etc. Thus, wavelength calibration is generally used for the spectroscopy, such as the light from a calibration lamp, tunable lasers, etc [12–16]. However, the conventional method is very difficult and costly [17]. Additional, characterization the spectral resolution of spectroscopic systems is also a difficult task. The high resolution measurement is needed for high resolution applications such as spectral interferometry and optical coherence tomography [18,19].
In this paper, we introduce a method for spectroscopic interferometry based on a rotating diffraction grating for measuring length ranges up to 4.0305 mm. Unlike conventional spectral interferometry, our proposed method increases the spectroscopic resolution by including the rotating diffraction grating. In addition, the effective measurement range can be extended considerably by including two diffraction gratings. Instead of calibrating the wavelength, we use the Fabry−Perot Etalon to calibrate the system. The unknown range is calculated from the peak position of the Fabry−Perot Etalon (known reference distance) using the fast Fourier transform. The reference period time (between two zero-order radiation) can be used to compensate the variation of the motor. Despite the simplicity of the proposed method, the variation of the rotation motor decreased considerably after compensation.
Therefore, the proposed method is optimized to handle and suitable for industrial use. It aims at cost minimization and can maintain high-speed measurement with a wide measuring range. Instead of calibrating the wavelength, we use the Fabry−Perot Etalon as standard to calibrate the system.
2. Principle
2.1 General optical spectrometer with diffraction grating
A diffraction grating is an optical component that splits and diffracts light into several beams traveling in different directions. As shown in Fig. 1, the light is collimated by a collimator toward the diffraction grating. The diffracted beam is then focused by a lens onto a line CCD for the spectral component detection. The directions of these beams depend on the spacing of the grating and the wavelength of the light; thus, the grating acts as a dispersive element. The relationship between the spacing of the grating and the wavelength is expressed as
where d is the distance from the center of one slit to the center of the adjacent slit, θi is the angle of incidence, θm is the angle between the diffracted ray and the grating's normal vector, m is an integer representing the propagation-mode of interest, and λ is the wavelength.
Fig. 1 Optical spectrometer with diffraction grating: (a) Schematic of optical spectrometer (b) Schematic of diffraction grating.
The line CCD is divided into a number of small, light-sensitive areas (known as pixels), which can be used to build up an image. The typical size of the CCD for infrared light is 1024 pixels, and the resolution of the CCD line camera is limited by the number of pixels, which in turn severely limits the length of pulses that can be measured with the spectrometer [20].
The calibration of the spectrometer is important due to the wavelength dispersion errors, and wavelength calibration is generally used for the grating spectrometer. However, this method is very difficult and costly [21]. Additional, characterization the spectral resolution of spectroscopic systems is also a difficult task.
2.2 Spectral interferometers using rotating diffraction grating
2.2.1 Rotating method of spectral interferometers
To circumvent the limitation of the CCD resolution, we proposed a spectral interferometer that includes a rotating diffraction device and a point photo detector. Thus, instead of increasing the number of pixels in the CCD, we used the rotating diffraction device to minimize the cost and improve the measuring speed.
Figure 2 illustrates the principle of operation of the proposed spectrometer. The beam from a low-coherence light source is collimated by lens collimator 1. The distance measurement is carried out by a Fizeau-type spectroscopic interferometer that includes a rotating diffraction grating, beam splitter, and circulator. The interfering beams pass through lens collimator 2 toward the rotating diffraction grating that deflects the dispersed beam toward the point photo detector for the spectral component measurements. The measuring speed can be improved considerably with a DC motor.
Fig. 2 Schematic showing the measurement principle of the proposed spectroscopic interferometer: (a) Optical path (b) Rotating diffraction device.
We assume that a single pulse in the Gaussian model emitted by the low-coherence light source is E0 (t), and the reference and measurement pulses are denoted by Eref (t) and Emeas (t), respectively [22]. Then, we use Fourier transformation to convert E (t) to E (ω), thus giving the spectrums of the reference and measurement pulses as
andrespectively, where and are the power factors showing the power difference of the pulse trains. The spectrums of the reference and measurement pulses with a time delay () are detected by the photo detector. The measured spectral intensity is expressed asFrom Eqs. (2) and (3), Eq. (4) is expanded to
where x is the unknown distance, and φ(ω)x represents the spectral phase position of the pulse trains, which is calculated from the peak position of the frequency using the fast Fourier transform.2.2.2 Rotating method with two diffraction gratings
We examined the prospect of extending the measurement range of the spectral interferometer by inserting a fixed diffraction grating in front of the rotating diffraction grating, as shown in Fig. 3. Collimators 1 and 2 operate as described previously in Section 2.2.1. The reconfigured Fizeau-type spectroscopic interferometer includes the two diffraction gratings, a measuring mirror, and a circulator. The interfering beams pass through lens collimator 2 through diffraction grating 1 (fixed) to diffraction grating 2 (rotating). The diffraction device is aligned in such a way that the first diffraction order beam retraces the incoming beam back through the fixed diffraction grating. The rotating diffraction grating deflects the dispersed beam towards the photo detector for the spectral component measurements.
Fig. 3 Optical layout of the proposed spectral interferometer with two diffraction gratings: (a) the light path of the diffraction device (b) the variation in the amplitude of the spectrum.
Figure 3(b) shows the variation in the amplitude of the spectrum passing though the diffraction gratings. The effective measuring range of the spectral interferometer is extended considerably by using two diffraction gratings.
Resolvance (R) for a device used to separate the wavelengths of light is defined as
where is the smallest resolvable wavelength difference. The limit of resolution is determined by the Rayleigh criterion as applied to the diffraction maxima, i.e., two wavelengths are just resolved when the maximum of one lies at the first minimum of the other. Since the space between maxima for slits is broken up into subsidiary maxima, the distance to the first mimimum is essentially times the separation of the main maxima. This leads to a resolvance for a grating ofwhere is the diffraction order, is the number of illuminated grooves over 1mm, is the diameter of the beam (mm) and is the total number of illuminated grooves.The measuring resolution is mainly affected by the hardware resolution of the anolog-to-digital conversion of the spectrometer and the computational resolution of the Fourier-transform software adopted for determining peak position. Therefore, the influence of aberration of the sensor may be one reason to the measuring resolution.
For spectral interferometer, the wavelength is determined from the relation of
where denotes the thickness and is the integral.where and are the wavelengths of the interference fringe. From Eqs. (9), the resolvable wavelength difference is calculated by a simple arithmetic:where is far larger than . From the Eq. (10), we note that the resolvable wavelength difference of the spectral interferometer is inversely proportional to the thickness , the maximum measurable distance can be expressed as:Here the number of illuminated grooves (over 1 mm) N is 600 and the order of the diffraction m is 1. The diameter of the beam W is 5 mm and the wavelength λ is 1550 nm. From the Eq. (7), we can calculate the theoretical value of the smallest resolvable wavelength difference Δλ is 0.52 nm. According to Eq. (11), the maximum measurable distance xd, xd <2.31 mm. Matsumoto conducted a series of experiments indicate that a length range of approximately 1.5 mm is feasible by using one diffraction grating [22]. The measurable resolvable of a grating is 0.80 nm. The measurable resolvable of two gratings is 0.40 nm. From the Eq. (7) and (11), the maximum measurable distance with two diffraction gratings is 4.62 mm.
3. Experiments and results
3.1 Configuration of the experiment
The photographs in Fig. 4 show our experimental setup. A superluminescent diode (SLD) (S5FC1005S, Thorlab) was used as the low-coherence light source. The center wavelength of the pulses was 1550 nm. As described in Section 2, the collimated beam of light arrives at the Fizeau-type spectroscopic interferometer, where one part is reflected by the beam splitter to form the reference arm, and the other is reflected by the measuring mirror and propagated to form the measurement arm. The interfering beams pass through lens collimator 2 toward the diffraction device. The diffraction device consists of a fixed diffraction grating (Edmund Optics #43748, nominal blaze wavelength 1600 nm, the number of illuminated grooves over 1 mm is 600, blaze angle 28° 41′), a rotating diffraction grating (Edmund Optics #43748), and a point photo detector (Thorlabs PDA10CS, DC-17 MHz, 900−1700 nm, 0.8 mm2). The rotating diffraction grating, which is rotated by a DC motor (Sawamura Denki SS23F), deflects the dispersed beam toward the detector for the spectral component measurements. Throughout the experiment, the temperature fluctuation in the laboratory was controlled to less than 0.1°C. The pressure and humidity were monitored with an accuracy of 1 hPa and 0.1%, respectively.
3.2 Measurement results
3.2.1 Calculating the angle peak frequency fa, from the time peak frequency ft and period time t
In the experiment, the reference period time (t) between two zero-order radiation is 0.1665 s, as shown in Fig. 5. The rotating speed of the DC motor (v) was 360 rpm, as calculated from the reference period time. When the rotating speed of the DC motor changed, we can use the reference period time (t) for correction.
Then, we obtain the spectrograms, as shown in Fig. 6(a). Last, by using the fast Fourier transform, we calculated the amplitude with the time peak frequency ft, as shown in Fig. 6(b). The angle peak frequency fa can be expressed as:
Fig. 6 Association with Fabry−Perot Etalon: (a) measured spectrograms (b) Fourier-transformed amplitude.
We note that the relationship between the length ranges and the original path length difference can be denoted as:
where a is the sensitivity of the angle peak frequency. Instead of calibrating the rotating speed, we use angle peak frequency fa to compensate the variation of the motor.3.2.2 Using Fabry−Perot Etalon to calculate the original path length difference (x0)
The experiment involved three steps. First, we used the Fabry−Perot Etalon to obtain the reference Fourier-transformed amplitude with the peak frequency (fr) of 149.4161 Hz, as shown in Fig. 6(b). Next, Fabry−Perot Etalon cancel, the measurement mirror was mounted on a small translation stage and positioned to give a relatively small (almost 1 mm) original path length difference (x0) between the reference and measurement arms. Last, by moving the mirror further away from the beam splitter in increments of 0.01 mm (measured by the laser hologage with displacement accuracy of 0.1 μm and repeatability of 0.02 μm), the separation between the two pulses increased and more fringes appeared in the measured spectrum. Using the fast Fourier transform, we calculated the amplitude of the interference spectrum accurately from x0 to x0 + 0.16 mm. Figure 7 shows the displacement for the measurement range of 0.16 mm in steps of 0.01 mm, and the calibration distance xoffset is 0.0688 mm. The free spectral range of Fabry−Perot Etalon is xr (150GHz correspond to 0.9993 mm). The original path length difference x0 was calculated from the peak frequency shown in Fig. 7(b) and the free spectral range of Fabry−Perot Etalon as x0 = xr − xoffset, which resulted in a value of 0.9305 mm.
Fig. 7 Displacement for the measurement range of 0.11 mm in steps of 0.01 mm: the original path length difference (x0) is calculated from the peak frequency of the Fabry−Perot Etalon.
In our experiment, instead of calibrating the wavelength, we use the Fabry−Perot Etalon (150GHz) to calibrate the system. The unknown range is calculated from the peak position of the Fabry−Perot Etalon (known reference distance) using the fast Fourier transform. During the measurement, the high-precision displacement measuring machine (laser hologage, 0.1 μm displacement accuracy, 0.02 μm repeatability, 0.01 μm resolution) is used for evaluation of the system.
3.2.3 Experiment for maximum measurable length
The measurement mirror was set such that the path length difference between the reference and measurement arms was 0.8305 mm. Then, by moving the mirror further away from the beam splitter in increments of 0.2 mm, the separation between the two pulses increased (from 0.8305 mm to 4.6305) and more fringes appeared in the measured spectrum, as shown in Fig. 8. This process was repeated three times independently. Using the fast Fourier transform, we calculated the amplitude of the interference spectrum and obtained the results shown in Fig. 9. The peak frequency increases with the path length difference. However, this relationship changes slightly at the higher values of the path length difference because the number of fringes in the measured spectrum exceeds the resolution of the spectrometer. Our experiments indicate that a length range of approximately 4.0 mm.
Fig. 8 Measured spectrograms with different length ranges (x): (a) 0.8305, (b) 1.0305, (c) 1.2305, (d) 1.4305, (e) 1.6305, (f) 1.8305, (g) 2.0305, (h) 2.2305, (i) 2.4305, (j) 2.6305, (k) 2.8305, (l) 3.0305, (m) 3.2305, (n) 3.4305, (0) 3.6305, (p) 3.8305, (q) 4.0305, (e) 4.2305, (s) 4.4305, and (t) 4.6305 mm.
Fig. 9 Fourier-transformed amplitude of the interference spectrum for different length ranges (x) corresponding to the values in Fig. 8.
The angle peak frequencies are plotted against the distance x in Fig. 10; Fig. 10 shows a linearity in the wide range with variation around several μm. The peak frequency increases with the path length difference. However, this relationship changes slightly at the higher values of the path length difference because the number of fringes in the measured spectrum exceeds the resolution of the spectrometer. The high number of fringes causes the modulation depth to become shallower, and the signal-to-noise ratio decreases, thus generating errors in both the peak frequency and the measuring length range [23]. In the future works, we also plan to analyze factors influencing measurement accuracy for high-precision measurement.
3.3. Verification repeatability and absolute precision
In order to verify the standard deviation of the measurement taken by rotating diffraction grating in the real environment, we measured the variation of rotary stage for ten times in increments of 0.5 mm. Here, , represents the standard deviation and average of the angle peak frequency . The standard deviation of the distance can be expressed as:
Figure 11 shows the results of the variation in the wide range. From the calibrated results, as shown in Table 1, we note that use angle peak frequency fa for compensation that the proposed setup to be insensitive to environmental vibration. Our experiments indicate that a length range of approximately 4.00 mm with repeatability for the narrow range were 0.17 μm (0.016%) and 3.84 μm (0.095%) for the wide range.
Table 1. Comparison results in the wide range
4. Conclusion
We proposed and developed a new scheme for dispersive interferometry using rotating diffraction grating. By using the rotation diffraction grating and point photo detector, the cost can be decreased greatly compared with the infrared CCD line camera. Moreover, the effective measuring range is extended considerably by using two diffraction gratings. This scheme is simple, aiming to minimize the cost, can maintain high-speed measurement with a wide measuring range, using the Fabry−Perot Etalon to calibrate the system, suitable for industrial use.
In our experiment, instead of calibrating the wavelength, we used the Fabry−Perot Etalon to calibrate the system. We calculated the ratio of the period times of the reference and measurement pulses based on the spectrometer results and calibrated the system using this ratio and the reference distance, which is the free spectral range of the Fabry−Perot Etalon. The unknown range was calculated from the peak position of the Fabry−Perot Etalon using the fast Fourier transform. During the measurement, the high-precision displacement measuring machine (laser hologage) is used for evaluation of the system. Our experiments demonstrate that the spectroscopic interferometer with rotating diffraction grating can measure length ranges up to 4.0305 mm, which indicates the feasibility of this scheme for industrial use. We observed that when the number of fringes exceeded the resolution of the spectrometer, the modulation depth became shallower and the signal-to-noise ratio decreased. These changes resulted in errors in the peak frequency and limited the maximum length range that was measured. The peak frequency and speed of the motor can be used to compensate the variation of the motor. Despite the simplicity of the proposed calibration method, the variation of the motor decreased considerably with compensation. Repeatability for the narrow range were 0.17μm (0.0167%) and 3.84 μm (0.0955%) for the wide range.
This method may also be applied to high-speed measurements by increasing the rotating speed of the DC motor. In the future work, our proposed method will be reliable in for absolute distance measurement with large length range.
Funding
National Natural Science Foundation of China (61803093, 61805048, 51305084, 61727810, 61801127, 61672168, 61803100). Science and Technology Project of Guangdong Province (2017B090911012), University Innovation and Entrepreneurship Education Major Project of Guangzhou City (201709P05).
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