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Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement
Optics Communications 382 (2017) 604–609
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement Dong Wei n, Masato Aketagawa Department of Mechanical Engineering, Nagaoka University of Technology, Nagaoka City, Niigata 940-2188, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 May 2016 Received in revised form 12 August 2016 Accepted 18 August 2016 Available online 26 August 2016
In this study, we attempt the separation of overlapped interference fringes arising from multiple pulse trains of a femtosecond optical frequency comb for length measurement. Based on an optical experiment, we test the performance of the separation of two overlapped interference fringes by time division for an absolute length measurement, which is about one adjacent pulse repetition interval length. We compare our results with those of a commercial He–Ne interferometer system. The two sets of results show an agreement within 0.7 mm. & 2016 Elsevier B.V. All rights reserved.
Keywords: Adjacent pulse repetition interval length Pulse repetition frequency Frequency comb Length measurement Metrology
1. Introduction The meter, which is the standard unit of length, is defined by the speed of light c in vacuum. One means of practical realization of the definition of the meter is in terms of wavelength via optical frequency standards (e.g., [1]). This approach exploits the relationship c = λ × f that relates the wavelength λ and frequency f , because the speed of light in vacuum c is a constant. Consequently, stabilizing the frequency of a laser can stabilize its wavelength. As regards length measurement, the displacement corresponding to a wavelength can be measured by means of a Michelson interferometer. In the above backdrop, femtosecond optical frequency combs (FOFCs) exhibiting high frequency stability have recently been reported (e.g., [2]). The FOFC is a representation of multiple phasecoherent frequencies. These frequencies are ranged with the same frequency interval, i.e., the pulse repetition frequency frep. The pulse repetition frequency is a parameter that needs to be stabilized to obtain a highly frequency-stabilized FOFC. The pulse repetition frequency and the adjacent pulse repetition interval length (APRIL, which represents the physical length associated with the pulse repetition period), Λ, are related as c = Λ × frep . Thus, stabilization of the pulse repetition frequency indicates APRIL stabilization in the spatial domain. In addition, the n
Corresponding author. E-mail address: [email protected] (D. Wei).
http://dx.doi.org/10.1016/j.optcom.2016.08.042 0030-4018/& 2016 Elsevier B.V. All rights reserved.
displacement corresponding to APRIL can be measured with a Michelson interferometer system [3–5]. From the detection possibilities of a Michelson interferometer and the stability derived from the frequency parameter, we have previously proposed the practical realization of the meter in terms of APRIL via the repetition frequency (e.g., [6–8]). We have also previously proposed multiple-pulse-train interferometry for arbitrary length measurements [9]. The proposed multiple-pulse-train interferometer has two mirrors in its object arm: one is denoted the zero-position mirror that indicates the zero-position of the measurement while the second is a target mirror that indicates the measurement point. As can be seen in the principles section in this paper, the problem with this setup is that we cannot distinguish the interference fringes of pulse trains reflected by the zero-position mirror and the target mirror when the reflected pulse trains are temporally and spatially overlapped. Thus, in this study, we propose the separation of these two interference fringes by time division for length measurement. Further, we compare our measurement result with that of a commercial He–Ne interferometer system. This paper is organized as follows. First, the problem with APRIL-based length measurement is highlighted in Section 2. The optical experiments for length measurement with the use of time division, the corresponding results, and error analysis are described in Section 3. Finally, the main conclusions are provided in Section 4.
D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
2. Methods By changing the light source of a Michelson interferometer to an FOFC, we can only measure a length that is equal to an integral multiple of the APRIL [3]. We note here that different pulse trains can interfere with each other only when they overlap. Consequently, to solve this problem, we propose the approach of multiple-pulse-train interferometry [9]. The difference between the multiple-pulse-train interferometer and the conventional Michelson interferometer lies in the object arm. As shown in Fig. 1(a), in a conventional Michelson interferometer, there is only one mirror in the object arm. The movement of this mirror induces a displacement that we want to measure. On the other hand, as shown in Fig. 1(b), the proposed multiple-pulse-train interferometer has two mirrors in the object arm. One mirror is the object mirror that indicates the zero-position of the measurement, and it is referred to as the zero-position mirror. The second mirror is an object mirror at the measurement point, and it is labeled the target mirror. The distance between these two mirrors is the distance that we want to measure. When representing the length in terms of APRIL, an arbitrary absolute length can be represented by an integer part, p, and a fractional part, q, of the APRIL of choice. After obtaining the integer and fractional parts, the total length can be obtained as L = (p + q ) × Λ . As shown in Fig. 2(a), when the fractional part, q, is large, two interference fringes are observed: one is formed between the reference mirror and the zero-position mirror, and the other is formed between the reference mirror and the target mirror. These two fringes are clearly separated. In this case, we can measure the fractional part based on the fringe analysis method [9,10]. On the other hand, when the fractional part is small (as shown in Fig. 2 (c)), while we observe interference phenomena between the two interference fringes [11], we cannot detect the envelope peak of the overlapped interference fringes. Therefore, it is impossible to perform a length measurement. Fig. 2(b) and (d) show examples of experimentally observed interference fringes. To measure small values of the fractional part, we propose to separate these two interference fringes by using a time division approach for length measurement. It is possible to avoid the superposition of both the interference fringes by moving the zeroposition mirror to introduce a spatial division at the expense of the introduction of a position error of the zero-point. The idea of time division is simple. The problem here is that
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two interference fringes caused by different pulse trains overlap temporally and spatially. Thus, we open or close a shutter in the path of the light beam to control the presence or absence of reflected light. For example, to select the reflected light from the target mirror, we open the shutter in the path of the target mirror. At the same time, we close the shutter in the path of the zeroposition mirror to block the reflected light from the zero-position mirror.
3. Optical experiments and results We first discuss the selection of the optical scheme for the realization of time division. Fig. 3 shows two possible amplitude-division approaches to achieve time division. In Fig. 3(a) and (b), the relay mirror is “cut” in or out of the beam to achieve time division by changing the direction of light. In Fig. 3(c) and (d), the relay mirror partially obscures the beam path, following [12]. The relay mirror bends half of the light perpendicular to the original traveling direction. Further, the reflected light is selected by opening or closing the appropriate shutter. In our study, we utilized the second scheme for our optical experiments, since this scheme does not require position control of the relay mirror. In preliminary experiments, we repeatedly set the relay mirror to a certain position. The reproducibility of the position of the relay mirror was about 1 mm. On the other hand, with the first scheme, the position of the reference mirror changes for each measurement, which can severely affect the measurement accuracy of the zero-position mirror. We next describe our optical experiments. Our setup consists of an FOFC-based interferometer system and commercial He–Ne interferometer system (Agilent, 10,766 A). The schematic of the experiment is shown in Fig. 4. First, we present the FOFC interferometer system. We first discuss the FOFC light source utilized in our setup. By tracing the frequency signal of the Global Positioning System, we use a frequency standard (pendulum, GPS-12R) to generate a signal of 10 MHz with a frequency stability on the order of 10–11. This stable 10-MHz signal is sent to a high-frequency signal generator (Digital Signal Technology, DPL-3.2GXF), which generates a frequency close to the pulse repetition frequency of the FOFC. The pulse repetition frequency of the FOFC is stabilized by locking the pulse repetition frequency of the FOFC to the stable frequency
Fig. 1. Schematic of (a) Michelson interferometer and (b) multiple-pulse-train interferometer.
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D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
Fig. 2. Formation of interference fringes. (a) Schematic of two separable interference fringes. (b) Observed interference fringes for case (a). (c) Schematic of two inseparable interference fringes. (d) Observed interference fringes for case (c).
Fig. 3. Schematic to realize time division. (a,b) All-pass all-bending system. (a) Selection of reflected light from target mirror. (b) Selection of reflected light from zeroposition mirror. (c,d) Half-pass half-bending system. (c) Selection of reflected light from target mirror. (d) Selection of reflected light from zero-position mirror.
generated by the high-frequency signal generator. In our study, the stability of the pulse repetition frequency of the FOFC was confirmed with the use of a frequency counter (pendulum, CNT-90). The Allan variance of the pulse repetition frequency was found to be less than 10 9 or more. During the optical experiment, the
pulse repetition frequency was measured by the same frequency counter. The power of the FOFC laser used in the study was greater than 3 mW, and the central wavelength and spectrum width were 1562 nm and 22 nm, respectively. The diameter of the beam was 1.6 mm, which was determined by the size of the collimator lens.
D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
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Fig. 4. Schematic of optical experiment.
As regards the FOFC system, to ensure linear movement of the target mirror, the target mirror was moved on a rail. The length of the rail was limited to the size of the “optical” table used for the setup. In the experiment, we used a rail with a length of 600 mm. Further, the pulse repetition frequency and half of the APRIL were about 70.616 MHz and 2.12 m, respectively. Therefore, we changed the length of APRIL by using a fiber etalon to enable measurements of around one APRIL. The output of the FOFC was transmitted to a fiber etalon to modify the repetition frequency. By using a fiber etalon to select the frequency of the comb, it is possible to obtain integer multiples of the pulse repetition frequency of the FOFC (e.g., [13,14]). In this experiment, we used a fiber etalon to change the pulse repetition frequency from frep to f ′rep = 4 × frep . Therefore, APRIL reduced from Λ to Λ′ = Λ/4 ≈ 530 mm . The pulse train emitted from the etalon was made incident on the beam splitter to generate interference fringes. We next describe the use of the piezoelectric transducer (PZT) scanner affixed to the reference mirror. A sawtooth wave signal generated by a signal generator was used to drive the PZT scanner. The PZT was used to vary the relative optical path difference between the two mirrors (in Fig. 4, these mirrors are the target and reference mirrors). A phase difference was obtained between the pulses reflected by the two mirrors, and interference fringes were observed. More detailed information about the formation of interference fringes can be found in Ref. [15]. Since the PZT is used for the observing interference fringes, its movement distance must cover the full width of the interference fringe. The amount of movement of the PZT was 100 mm in this experiment. The interference fringes of the FOFC and length data of He–Ne interferometer system were obtained as follows. First, we describe the acquisition of fringes with the FOFC-based interferometer. The target mirror was set at a position close to the He–Ne interferometer system. At this time, we selected the reflected light from the zero-position mirror. The interference fringes of the zero-position mirror and the reference mirror were recorded. Next, by opening and closing of the appropriate shutters, we selected the reflected light from the target mirror. The interference fringes of the reference mirror and the target mirror were subsequently recorded. Next, the target mirror was moved along the rail to a position close to the FOFC. Subsequently, the interference fringes of the reference mirror and the zero-position mirror and the interference pattern of the reference mirror and target mirror were recorded. During the experiment, each of the interference fringes was recorded 10 times. The same measurement was performed four times.
Next, we describe the acquisition of displacement data with the use of the He–Ne interferometer system. As part of the study design, we planned to compare the length measurement results obtained using the FOFC with those of a He–Ne interferometer system. Here, we mention that the He–Ne interferometer system can only measure displacement. The commercial He–Ne interferometer was setup according to its manual. The measurement results obtained with this interferometer were recorded by a personal computer. We recall here that the object mirror was initially set at a position close to the He–Ne interferometer system. The length readout value of the He–Ne interferometer system was set to zero. Experimental data obtained with the He–Ne interferometer system were continuously recorded until all the interference fringes of the FOFC-based interferometer were acquired. During the experiment, temperature (T), barometric pressure (P), and humidity (H) were recorded by a thermometer (Testo, 735), barometer (Sunoh, VR-18), and hygrometer (VAISALA, HM70), respectively. Data processing was performed as follows. The movement of the object mirror was evaluated by the APRIL of the FOFC and the He–Ne interferometer system. The length value measured by APRIL was obtained as follows. When the object mirror was set to a fixed position (close to the He–Ne interferometer system or the FOFC), we recorded two interference fringes. One was used to confirm the zero-position of the measurement, and the other fringe was used to determine the relative position of the object mirror. From the difference between the two interference fringes, we evaluated the two amounts of deviations of the object mirror from the zero reference position. The moving distances were set close to one APRIL. With the use of the two obtained deviations to correct the amount of movement from the value of one APRIL, the displacement of the object mirror was measured by using APRIL. The value of APRIL was calculated based on the pulse repetition frequency measured by the frequency counter. Fig. 5 shows an example of the length measurement. We next describe the displacement measured by the He–Ne interferometer system. During the experiment, the positions of the object mirror were recorded by the He–Ne interferometer system. While acquiring the interference fringes of the FOFC, the measured value of the He–Ne interferometer system fluctuated due to the vibration of the optical table and the fluctuation of air. Thus, we used the averaged measurement value of the He–Ne interferometer system. The average time was equal to the acquisition time required to obtain 10 interference fringes of the FOFC when
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D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
Fig. 5. (a) Target mirror set close to the FOFC side with fractional part Δ1 = q1 × Λ . (b) The movement of the target is equal to APRIL, and thus, Δ2 = Δ1 = q1 × Λ . (c) The movement of the target is larger than APRIL, and thus, Δ2 > Δ1 = q1 × Λ . (d) The movement of the target is smaller than APRIL, and thus, Δ2 < Δ1 = q1 × Λ .
the zero-position or target mirror were at a fixed position. The measured air temperature, air pressure, and humidity data were used to calculate the phase refractive and group refractive indices of air. The phase refractive index of air was calculated based on the Edlén equations [16–18]. The group refractive index of air was calculated as per the definition provided in Ref. [19]. The lengths measured by APRIL and the wavelength were converted to lengths in vacuum with the use of the calculated refractive indices. Both sets of lengths were finally compared for evaluation. Fig. 6 compares the results of the target mirror movement measured by the FOFC-based and He–Ne interferometer systems. From Fig. 6, we note that there is a maximum deviation of 0.7 mm between the two average values. The bars for all measurements in Fig. 6 are 1 mm width, which is equal to the 2s value of the standard deviation of the measured results. Here, we recall that each of the interference fringes was recorded 10 times for each measurement. Based on these fringes, we calculated the average value of the position of the target mirror or the zero-position mirror. The observed variation in measurements was due to the movement and fixing of the target mirror on the rail, which were performed manually. During the mirror movement, the orientation of the mirror also changed. The actual movement length (which was evaluated by the He–Ne interferometer system) was different from the linear distance between
Fig. 6. Difference between the measured values obtained with both methods.
the starting point and end point of the measurement (which was evaluated by the FOFC interferometer system). During each measurement instant, the fixed position (the starting point or end point) of the target mirror was different. The orientation of the target mirror at each fixed location was also different. Depending on the direction of the target mirror, the linear distance between the starting point and end point also slightly changed every time. We next performed the following error analysis. We first mention that the factors of environmental parameters (temperature, pressure, humidity, etc.) variation, mechanical vibration, nonlinearity of the PZT actuator, method of detection of the interference fringe peak, and so on are important for evaluation of the uncertainty of measurement. During the experiment, changes in the temperature, pressure, and humidity (which are defined as the absolute difference value between the maximum and minimum values) were 0.07 °C, 0.4 hPa, and 0.5%, respectively. The corresponding values of the sensitivity coefficients of the refractive index of air were known to be 1 ppm/1 °C,0.27 ppm/1 hPa, and 0.007 ppm/1% (for example, see [8]). Changes in the measured values due to variations in the environmental parameters were calculated to be 40 nm (E0.07 °C 1 ppm/1 °C 0.53 m E 40 nm), 40 nm (E 0.4 hPa 0.27 ppm/1 hPa 0.53 m E40 nm), and 3 nm (E 0.5% 0.007 ppm/1% 0.53 m E 3 nm). The nonlinearity of the PZT actuator was measured with the He–Ne interferometer system. The residual nonlinearity error of the PZT actuator was estimated to be less than 100 nm. The error in the peak detection method of the interference fringes was estimated to be less than 50 nm (for example, see [20]). The order of uncertainty of the measurement results is smaller than the order of the uncertainties of the abovementioned errors by at least one digit. Consequently, we speculate that the Abbe error forms a major error factor with our setup. When the optical axes of the FOFC-based interferometerand He–Ne interferometer system do not lie in a straight line, Abbe error occurs. The Abbe error, δabbe , can be expressed as
δabbe =
δpitch2 + δyaw 2 .
The
angular
changes
due
to
the
D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
displacement of the object mirror were measured with an autocollimator (Chuo Precision Industrial, LAC). The average values of the measured pitching and yawing, θpitch and θyaw , were 0.4 and 0.9 mrad, respectively. These variations in pitching and yawing (defined here as the absolute difference value between the maximum and minimum values), were 0.019 mrad and 0.02 mrad, respectively. The variation in pitching and yawing forms a significant contribution to the difference between each measurement result, as shown in Fig. 6. This pitch-and-yaw variation arose from the manual operation (movement and fixing) of the target mirror on the rail. Further, we defined d as the amount of mismatch between the optical axes of the FOFC-based interferometer and He– Ne interferometer system, which was estimated to be about 0.7 mm. We estimated the value of d with the naked eye based on the spot diameter of the beam of the He–Ne laser. Substituting these values into the above equation, the Abbe error was estimated to be 0.70 mm, as shown below. δpitch ≈ d × θpitch ≈ 0.30 μm, δyaw ≈ d × θyaw ≈ 0.63 μm, and
δabbe = δpitch2 + δyaw 2 ≈ 0.70 μm. Here, we mention that we plan to undertake a more detailed examination on the uncertainty of measurement in our future studies; at present, we are concerned only with the feasibility of the proposed method. In summary, we confirmed the successful application of the time division approach to separate two temporally and spatially overlapped interference fringes. The proposed method is significant in the sense that we were able to measure lengths that cannot be measured by conventional methods.
4. Conclusion When using multiple-pulse-train interferometers for length measurements, we cannot distinguish the interference fringes of the pulse trains reflected by the zero-position mirror and the object mirror when the fringes temporally and spatially overlap. In this work, we separated the two interference fringes by means of time division. We utilized the separated interference fringes to perform absolute length measurements that were about one APRIL long. Our measurement results were compared with those of a commercial He–Ne interferometer system. The difference between the averaged values of the two systems was 0.7 mm. The main error factor was the Abbe error due to the mismatch of optical axes of the two systems. In our future studies, we plan to use a light source with a variable APRIL and to introduce an integer-part estimation unit in the interferometer. With these two improvements, we aim to perform arbitrary absolute length measurements with an FOFC.
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Acknowledgments We thank Mr. Yasuhiro Sakai of Nagaoka University of Technology, who performed the measurements. This research work was partially financially supported by research grants from the Mazda Foundation (No. 2013-8) and the Support Centre for Advance Telecommunications Technology Research (SCAT) (No. H26-1).
References [1] T.J. Quinn, Practical realization of the definition of the metre, including recommended radiations of other optical frequency standards (2001), Metrologia 40 (2003) 103. [2] J. Ye, S.T. Cundiff, Femtosecond Optical Frequency Comb : Principle, Operation, and Applications, Springer, New York, NY, 2005. [3] J. Ye, Absolute measurement of a long, arbitrary distance to less than an optical fringe, Opt. Lett. 29 (2004) 1153–1155. [4] M. Cui, R.N. Schouten, N. Bhattacharya, S.A. Berg, Experimental demonstration of distance measurement with a femtosecond frequency comb laser, J. Eur. Opt. Soc. Rapid Publ. 3 (2008) 08003. [5] D. Wei, K. Takamasu, H. Matsumoto, A study of the possibility of using an adjacent pulse repetition interval length as a scale using a helium–neon interferometer, Precis. Eng. 37 (2013) 694–698. [6] D. Wei, M. Aketagawa, Comparison of length measurements provided by a femtosecond optical frequency comb, Opt. Express 22 (2014) 7040–7045. [7] D. Wei, M. Aketagawa, Characteristics of an adjacent pulse repetition interval length as a scale for length, Opt. Eng. 53 (2013), 051502-051502. [8] D. Wei, M. Aketagawa, Uncertainty in length conversion due to change of sensitivity coefficients of refractive index, Opt. Commun. 345 (2015) 67–70. [9] D. Wei, S. Takahashi, K. Takamasu, H. Matsumoto, Time-of-flight method using multiple pulse train interference as a time recorder, Opt. Express 19 (2011) 4881–4889. [10] P. Rastogi, E. Hack, Phase Estimation in Optical Interferometry, CRC Press, 2014. [11] D. Wei, S. Takahashi, K. Takamasu, H. Matsumoto, Experimental observation of pulse trains' destructive interference with a femtosecond optical frequencycomb-based interferometer, Opt. Lett. 34 (2009) 2775–2777. [12] G. Wu, K. Arai, M. Takahashi, H. Inaba, K. Minoshima, High-accuracy correction of air refractive index by using two-color heterodyne interferometry of optical frequency combs, Meas. Sci. Technol. 24 (2013) 015203. [13] C. Narin, T. Satoru, T. Kiyoshi, M. Hirokazu, A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser, Meas. Sci. Technol. 23 (2012) 054003. [14] W. Sudatham, H. Matsumoto, S. Takahashi, K. Takamasu, Verification of the positioning accuracy of industrial coordinate measuring machine using optical-comb pulsed interferometer with a rough metal ball target, Precis. Eng. (2015) 63–67. [15] D. Wei, S. Takahashi, K. Takamasu, H. Matsumoto, Analysis of the temporal coherence function of a femtosecond optical frequency comb, Opt. Express 17 (2009) 7011–7018. [16] E. Bengt, The Refractive Index of Air, Metrologia 2 (1966) 71. [17] K.P. Birch, M.J. Downs, An updated Edlén equation for the refractive index of air, Metrologia 30 (1993) 155. [18] J.A. Stone, J.H. Zimmerman, Refractive index of air calculator. 〈http://emtool box.nist.gov/Wavelength/Edlen.asp〉. [19] B.E.A. Saleh, M.C. Teich, Fundamentals of photonics, Wiley Series In Pure and Applied Optics, Wiley-Interscience, Hoboken, N.J., 2007. [20] D. Wei, H. Matsumoto, Measurement accuracy of the pulse repetition intervalbased excess fraction (PRIEF) method: an analogy-based theoretical analysis, J. Eur. Opt. Soc. Rapid Publ. 7 (2012) 12050.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement Dong Wei n, Masato Aketagawa Department of Mechanical Engineering, Nagaoka University of Technology, Nagaoka City, Niigata 940-2188, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 30 May 2016 Received in revised form 12 August 2016 Accepted 18 August 2016 Available online 26 August 2016
In this study, we attempt the separation of overlapped interference fringes arising from multiple pulse trains of a femtosecond optical frequency comb for length measurement. Based on an optical experiment, we test the performance of the separation of two overlapped interference fringes by time division for an absolute length measurement, which is about one adjacent pulse repetition interval length. We compare our results with those of a commercial He–Ne interferometer system. The two sets of results show an agreement within 0.7 mm. & 2016 Elsevier B.V. All rights reserved.
Keywords: Adjacent pulse repetition interval length Pulse repetition frequency Frequency comb Length measurement Metrology
1. Introduction The meter, which is the standard unit of length, is defined by the speed of light c in vacuum. One means of practical realization of the definition of the meter is in terms of wavelength via optical frequency standards (e.g., [1]). This approach exploits the relationship c = λ × f that relates the wavelength λ and frequency f , because the speed of light in vacuum c is a constant. Consequently, stabilizing the frequency of a laser can stabilize its wavelength. As regards length measurement, the displacement corresponding to a wavelength can be measured by means of a Michelson interferometer. In the above backdrop, femtosecond optical frequency combs (FOFCs) exhibiting high frequency stability have recently been reported (e.g., [2]). The FOFC is a representation of multiple phasecoherent frequencies. These frequencies are ranged with the same frequency interval, i.e., the pulse repetition frequency frep. The pulse repetition frequency is a parameter that needs to be stabilized to obtain a highly frequency-stabilized FOFC. The pulse repetition frequency and the adjacent pulse repetition interval length (APRIL, which represents the physical length associated with the pulse repetition period), Λ, are related as c = Λ × frep . Thus, stabilization of the pulse repetition frequency indicates APRIL stabilization in the spatial domain. In addition, the n
Corresponding author. E-mail address: [email protected] (D. Wei).
http://dx.doi.org/10.1016/j.optcom.2016.08.042 0030-4018/& 2016 Elsevier B.V. All rights reserved.
displacement corresponding to APRIL can be measured with a Michelson interferometer system [3–5]. From the detection possibilities of a Michelson interferometer and the stability derived from the frequency parameter, we have previously proposed the practical realization of the meter in terms of APRIL via the repetition frequency (e.g., [6–8]). We have also previously proposed multiple-pulse-train interferometry for arbitrary length measurements [9]. The proposed multiple-pulse-train interferometer has two mirrors in its object arm: one is denoted the zero-position mirror that indicates the zero-position of the measurement while the second is a target mirror that indicates the measurement point. As can be seen in the principles section in this paper, the problem with this setup is that we cannot distinguish the interference fringes of pulse trains reflected by the zero-position mirror and the target mirror when the reflected pulse trains are temporally and spatially overlapped. Thus, in this study, we propose the separation of these two interference fringes by time division for length measurement. Further, we compare our measurement result with that of a commercial He–Ne interferometer system. This paper is organized as follows. First, the problem with APRIL-based length measurement is highlighted in Section 2. The optical experiments for length measurement with the use of time division, the corresponding results, and error analysis are described in Section 3. Finally, the main conclusions are provided in Section 4.
D. Wei, M. Aketagawa / Optics Communications 382 (2017) 604–609
2. Methods By changing the light source of a Michelson interferometer to an FOFC, we can only measure a length that is equal to an integral multiple of the APRIL [3]. We note here that different pulse trains can interfere with each other only when they overlap. Consequently, to solve this problem, we propose the approach of multiple-pulse-train interferometry [9]. The difference between the multiple-pulse-train interferometer and the conventional Michelson interferometer lies in the object arm. As shown in Fig. 1(a), in a conventional Michelson interferometer, there is only one mirror in the object arm. The movement of this mirror induces a displacement that we want to measure. On the other hand, as shown in Fig. 1(b), the proposed multiple-pulse-train interferometer has two mirrors in the object arm. One mirror is the object mirror that indicates the zero-position of the measurement, and it is referred to as the zero-position mirror. The second mirror is an object mirror at the measurement point, and it is labeled the target mirror. The distance between these two mirrors is the distance that we want to measure. When representing the length in terms of APRIL, an arbitrary absolute length can be represented by an integer part, p, and a fractional part, q, of the APRIL of choice. After obtaining the integer and fractional parts, the total length can be obtained as L = (p + q ) × Λ . As shown in Fig. 2(a), when the fractional part, q, is large, two interference fringes are observed: one is formed between the reference mirror and the zero-position mirror, and the other is formed between the reference mirror and the target mirror. These two fringes are clearly separated. In this case, we can measure the fractional part based on the fringe analysis method [9,10]. On the other hand, when the fractional part is small (as shown in Fig. 2 (c)), while we observe interference phenomena between the two interference fringes [11], we cannot detect the envelope peak of the overlapped interference fringes. Therefore, it is impossible to perform a length measurement. Fig. 2(b) and (d) show examples of experimentally observed interference fringes. To measure small values of the fractional part, we propose to separate these two interference fringes by using a time division approach for length measurement. It is possible to avoid the superposition of both the interference fringes by moving the zeroposition mirror to introduce a spatial division at the expense of the introduction of a position error of the zero-point. The idea of time division is simple. The problem here is that
605
two interference fringes caused by different pulse trains overlap temporally and spatially. Thus, we open or close a shutter in the path of the light beam to control the presence or absence of reflected light. For example, to select the reflected light from the target mirror, we open the shutter in the path of the target mirror. At the same time, we close the shutter in the path of the zeroposition mirror to block the reflected light from the zero-position mirror.
3. Optical experiments and results We first discuss the selection of the optical scheme for the realization of time division. Fig. 3 shows two possible amplitude-division approaches to achieve time division. In Fig. 3(a) and (b), the relay mirror is “cut” in or out of the beam to achieve time division by changing the direction of light. In Fig. 3(c) and (d), the relay mirror partially obscures the beam path, following [12]. The relay mirror bends half of the light perpendicular to the original traveling direction. Further, the reflected light is selected by opening or closing the appropriate shutter. In our study, we utilized the second scheme for our optical experiments, since this scheme does not require position control of the relay mirror. In preliminary experiments, we repeatedly set the relay mirror to a certain position. The reproducibility of the position of the relay mirror was about 1 mm. On the other hand, with the first scheme, the position of the reference mirror changes for each measurement, which can severely affect the measurement accuracy of the zero-position mirror. We next describe our optical experiments. Our setup consists of an FOFC-based interferometer system and commercial He–Ne interferometer system (Agilent, 10,766 A). The schematic of the experiment is shown in Fig. 4. First, we present the FOFC interferometer system. We first discuss the FOFC light source utilized in our setup. By tracing the frequency signal of the Global Positioning System, we use a frequency standard (pendulum, GPS-12R) to generate a signal of 10 MHz with a frequency stability on the order of 10–11. This stable 10-MHz signal is sent to a high-frequency signal generator (Digital Signal Technology, DPL-3.2GXF), which generates a frequency close to the pulse repetition frequency of the FOFC. The pulse repetition frequency of the FOFC is stabilized by locking the pulse repetition frequency of the FOFC to the stable frequency
Fig. 1. Schematic of (a) Michelson interferometer and (b) multiple-pulse-train interferometer.
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Fig. 2. Formation of interference fringes. (a) Schematic of two separable interference fringes. (b) Observed interference fringes for case (a). (c) Schematic of two inseparable interference fringes. (d) Observed interference fringes for case (c).
Fig. 3. Schematic to realize time division. (a,b) All-pass all-bending system. (a) Selection of reflected light from target mirror. (b) Selection of reflected light from zeroposition mirror. (c,d) Half-pass half-bending system. (c) Selection of reflected light from target mirror. (d) Selection of reflected light from zero-position mirror.
generated by the high-frequency signal generator. In our study, the stability of the pulse repetition frequency of the FOFC was confirmed with the use of a frequency counter (pendulum, CNT-90). The Allan variance of the pulse repetition frequency was found to be less than 10 9 or more. During the optical experiment, the
pulse repetition frequency was measured by the same frequency counter. The power of the FOFC laser used in the study was greater than 3 mW, and the central wavelength and spectrum width were 1562 nm and 22 nm, respectively. The diameter of the beam was 1.6 mm, which was determined by the size of the collimator lens.
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Fig. 4. Schematic of optical experiment.
As regards the FOFC system, to ensure linear movement of the target mirror, the target mirror was moved on a rail. The length of the rail was limited to the size of the “optical” table used for the setup. In the experiment, we used a rail with a length of 600 mm. Further, the pulse repetition frequency and half of the APRIL were about 70.616 MHz and 2.12 m, respectively. Therefore, we changed the length of APRIL by using a fiber etalon to enable measurements of around one APRIL. The output of the FOFC was transmitted to a fiber etalon to modify the repetition frequency. By using a fiber etalon to select the frequency of the comb, it is possible to obtain integer multiples of the pulse repetition frequency of the FOFC (e.g., [13,14]). In this experiment, we used a fiber etalon to change the pulse repetition frequency from frep to f ′rep = 4 × frep . Therefore, APRIL reduced from Λ to Λ′ = Λ/4 ≈ 530 mm . The pulse train emitted from the etalon was made incident on the beam splitter to generate interference fringes. We next describe the use of the piezoelectric transducer (PZT) scanner affixed to the reference mirror. A sawtooth wave signal generated by a signal generator was used to drive the PZT scanner. The PZT was used to vary the relative optical path difference between the two mirrors (in Fig. 4, these mirrors are the target and reference mirrors). A phase difference was obtained between the pulses reflected by the two mirrors, and interference fringes were observed. More detailed information about the formation of interference fringes can be found in Ref. [15]. Since the PZT is used for the observing interference fringes, its movement distance must cover the full width of the interference fringe. The amount of movement of the PZT was 100 mm in this experiment. The interference fringes of the FOFC and length data of He–Ne interferometer system were obtained as follows. First, we describe the acquisition of fringes with the FOFC-based interferometer. The target mirror was set at a position close to the He–Ne interferometer system. At this time, we selected the reflected light from the zero-position mirror. The interference fringes of the zero-position mirror and the reference mirror were recorded. Next, by opening and closing of the appropriate shutters, we selected the reflected light from the target mirror. The interference fringes of the reference mirror and the target mirror were subsequently recorded. Next, the target mirror was moved along the rail to a position close to the FOFC. Subsequently, the interference fringes of the reference mirror and the zero-position mirror and the interference pattern of the reference mirror and target mirror were recorded. During the experiment, each of the interference fringes was recorded 10 times. The same measurement was performed four times.
Next, we describe the acquisition of displacement data with the use of the He–Ne interferometer system. As part of the study design, we planned to compare the length measurement results obtained using the FOFC with those of a He–Ne interferometer system. Here, we mention that the He–Ne interferometer system can only measure displacement. The commercial He–Ne interferometer was setup according to its manual. The measurement results obtained with this interferometer were recorded by a personal computer. We recall here that the object mirror was initially set at a position close to the He–Ne interferometer system. The length readout value of the He–Ne interferometer system was set to zero. Experimental data obtained with the He–Ne interferometer system were continuously recorded until all the interference fringes of the FOFC-based interferometer were acquired. During the experiment, temperature (T), barometric pressure (P), and humidity (H) were recorded by a thermometer (Testo, 735), barometer (Sunoh, VR-18), and hygrometer (VAISALA, HM70), respectively. Data processing was performed as follows. The movement of the object mirror was evaluated by the APRIL of the FOFC and the He–Ne interferometer system. The length value measured by APRIL was obtained as follows. When the object mirror was set to a fixed position (close to the He–Ne interferometer system or the FOFC), we recorded two interference fringes. One was used to confirm the zero-position of the measurement, and the other fringe was used to determine the relative position of the object mirror. From the difference between the two interference fringes, we evaluated the two amounts of deviations of the object mirror from the zero reference position. The moving distances were set close to one APRIL. With the use of the two obtained deviations to correct the amount of movement from the value of one APRIL, the displacement of the object mirror was measured by using APRIL. The value of APRIL was calculated based on the pulse repetition frequency measured by the frequency counter. Fig. 5 shows an example of the length measurement. We next describe the displacement measured by the He–Ne interferometer system. During the experiment, the positions of the object mirror were recorded by the He–Ne interferometer system. While acquiring the interference fringes of the FOFC, the measured value of the He–Ne interferometer system fluctuated due to the vibration of the optical table and the fluctuation of air. Thus, we used the averaged measurement value of the He–Ne interferometer system. The average time was equal to the acquisition time required to obtain 10 interference fringes of the FOFC when
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Fig. 5. (a) Target mirror set close to the FOFC side with fractional part Δ1 = q1 × Λ . (b) The movement of the target is equal to APRIL, and thus, Δ2 = Δ1 = q1 × Λ . (c) The movement of the target is larger than APRIL, and thus, Δ2 > Δ1 = q1 × Λ . (d) The movement of the target is smaller than APRIL, and thus, Δ2 < Δ1 = q1 × Λ .
the zero-position or target mirror were at a fixed position. The measured air temperature, air pressure, and humidity data were used to calculate the phase refractive and group refractive indices of air. The phase refractive index of air was calculated based on the Edlén equations [16–18]. The group refractive index of air was calculated as per the definition provided in Ref. [19]. The lengths measured by APRIL and the wavelength were converted to lengths in vacuum with the use of the calculated refractive indices. Both sets of lengths were finally compared for evaluation. Fig. 6 compares the results of the target mirror movement measured by the FOFC-based and He–Ne interferometer systems. From Fig. 6, we note that there is a maximum deviation of 0.7 mm between the two average values. The bars for all measurements in Fig. 6 are 1 mm width, which is equal to the 2s value of the standard deviation of the measured results. Here, we recall that each of the interference fringes was recorded 10 times for each measurement. Based on these fringes, we calculated the average value of the position of the target mirror or the zero-position mirror. The observed variation in measurements was due to the movement and fixing of the target mirror on the rail, which were performed manually. During the mirror movement, the orientation of the mirror also changed. The actual movement length (which was evaluated by the He–Ne interferometer system) was different from the linear distance between
Fig. 6. Difference between the measured values obtained with both methods.
the starting point and end point of the measurement (which was evaluated by the FOFC interferometer system). During each measurement instant, the fixed position (the starting point or end point) of the target mirror was different. The orientation of the target mirror at each fixed location was also different. Depending on the direction of the target mirror, the linear distance between the starting point and end point also slightly changed every time. We next performed the following error analysis. We first mention that the factors of environmental parameters (temperature, pressure, humidity, etc.) variation, mechanical vibration, nonlinearity of the PZT actuator, method of detection of the interference fringe peak, and so on are important for evaluation of the uncertainty of measurement. During the experiment, changes in the temperature, pressure, and humidity (which are defined as the absolute difference value between the maximum and minimum values) were 0.07 °C, 0.4 hPa, and 0.5%, respectively. The corresponding values of the sensitivity coefficients of the refractive index of air were known to be 1 ppm/1 °C,0.27 ppm/1 hPa, and 0.007 ppm/1% (for example, see [8]). Changes in the measured values due to variations in the environmental parameters were calculated to be 40 nm (E0.07 °C 1 ppm/1 °C 0.53 m E 40 nm), 40 nm (E 0.4 hPa 0.27 ppm/1 hPa 0.53 m E40 nm), and 3 nm (E 0.5% 0.007 ppm/1% 0.53 m E 3 nm). The nonlinearity of the PZT actuator was measured with the He–Ne interferometer system. The residual nonlinearity error of the PZT actuator was estimated to be less than 100 nm. The error in the peak detection method of the interference fringes was estimated to be less than 50 nm (for example, see [20]). The order of uncertainty of the measurement results is smaller than the order of the uncertainties of the abovementioned errors by at least one digit. Consequently, we speculate that the Abbe error forms a major error factor with our setup. When the optical axes of the FOFC-based interferometerand He–Ne interferometer system do not lie in a straight line, Abbe error occurs. The Abbe error, δabbe , can be expressed as
δabbe =
δpitch2 + δyaw 2 .
The
angular
changes
due
to
the
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displacement of the object mirror were measured with an autocollimator (Chuo Precision Industrial, LAC). The average values of the measured pitching and yawing, θpitch and θyaw , were 0.4 and 0.9 mrad, respectively. These variations in pitching and yawing (defined here as the absolute difference value between the maximum and minimum values), were 0.019 mrad and 0.02 mrad, respectively. The variation in pitching and yawing forms a significant contribution to the difference between each measurement result, as shown in Fig. 6. This pitch-and-yaw variation arose from the manual operation (movement and fixing) of the target mirror on the rail. Further, we defined d as the amount of mismatch between the optical axes of the FOFC-based interferometer and He– Ne interferometer system, which was estimated to be about 0.7 mm. We estimated the value of d with the naked eye based on the spot diameter of the beam of the He–Ne laser. Substituting these values into the above equation, the Abbe error was estimated to be 0.70 mm, as shown below. δpitch ≈ d × θpitch ≈ 0.30 μm, δyaw ≈ d × θyaw ≈ 0.63 μm, and
δabbe = δpitch2 + δyaw 2 ≈ 0.70 μm. Here, we mention that we plan to undertake a more detailed examination on the uncertainty of measurement in our future studies; at present, we are concerned only with the feasibility of the proposed method. In summary, we confirmed the successful application of the time division approach to separate two temporally and spatially overlapped interference fringes. The proposed method is significant in the sense that we were able to measure lengths that cannot be measured by conventional methods.
4. Conclusion When using multiple-pulse-train interferometers for length measurements, we cannot distinguish the interference fringes of the pulse trains reflected by the zero-position mirror and the object mirror when the fringes temporally and spatially overlap. In this work, we separated the two interference fringes by means of time division. We utilized the separated interference fringes to perform absolute length measurements that were about one APRIL long. Our measurement results were compared with those of a commercial He–Ne interferometer system. The difference between the averaged values of the two systems was 0.7 mm. The main error factor was the Abbe error due to the mismatch of optical axes of the two systems. In our future studies, we plan to use a light source with a variable APRIL and to introduce an integer-part estimation unit in the interferometer. With these two improvements, we aim to perform arbitrary absolute length measurements with an FOFC.
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Acknowledgments We thank Mr. Yasuhiro Sakai of Nagaoka University of Technology, who performed the measurements. This research work was partially financially supported by research grants from the Mazda Foundation (No. 2013-8) and the Support Centre for Advance Telecommunications Technology Research (SCAT) (No. H26-1).
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