Abstract

A three-spherical-mirror test method that uses a Fabry-Pérot (FP) cavity is proposed for a radius of curvature measurement system, especially for radii larger than 10 m. By using the three-spherical-mirror test with mode spacing measurement in an FP cavity, the local value of the radius of curvature of a mirror can be determined in situ and this mirror can then be used as the reference spherical mirror in a radius of curvature measurement system. We demonstrated determinations of radii of curvature of around 10 m using the three-spherical-mirror test with uncertainties of around 1.5 × 10−4 and then measured the radius of curvature of around 20 m with uncertainties of around 3.1 × 10−4 by using the spherical mirror, of which the radius of curvature was determined by the three-spherical-mirror test, as the reference sphere. The proposed system has high practical applicability because measurements can be conducted under usual air conditions and the measurement results are directly traceable to the time standard because beat frequency measurement is used.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

To construct or evaluate high-reliability optical devices and systems, such as compound lens, telescopes, computer-generated holograms, and scientific research equipment, highly accurate reference standards of the radius of curvature, such as Newton gauges, are important. Accurate measurement of the radius of curvature of a spherical surface is important for progress in engineering and science. For example, for gravitational wave antennas, the radius of curvature of the spherical mirror must be sufficiently accurate [1]. In the large precision optical systems, such as astronomical telescopes and laser fusion programs, high accurate measurement for a large radius of curvature is required for spherical components [2–4].

Interferometric systems that use a Fizeau-type phase-shifting spherical interferometer and a displacement-measuring interferometer are well established and practically applied for highly accurate measurement of the radius of curvature [5]. Systems that combine a confocal signal processing apparatus and a displacement-measuring interferometer have been proposed [6,7]. In these systems, the distance between the cat’s-eye and confocal positions is measured via mechanical scanning and the measured value corresponds to the average of the radius of curvature over the area illuminated by the laser beam. Although these systems have a relative uncertainty of better than 10−5 for radii less than 1 m, it is difficult to measure a radius of curvature larger than 1 m. To shorten the mechanical scanning distance for a large radius of curvature measurement, methods for folding the measurement beam in the interferometer have been proposed [8,9]. In addition, interferometric methods that utilize references of a large radius of curvature, including one in which the focal lengths of two lenses are calibrated [10] and one that uses holographic radius test plates [11], have been proposed. These methods can measure the radius of curvature for radii larger than 10 m and their relative uncertainty is on the order of 10−4.

The radius of curvature can be calculated from the directly measured surface profile of the target sphere. For high-accuracy measurement of small targets, a micro-coordinate measuring machine is often adopted [12]. For large targets, a laser tracker system that uses an interferometer has been developed [13]. The measurement accuracy of contact methods depends on the sagitta of the measured profile. A small sagitta, measured for targets with a large radius of curvature or small aperture, leads to large uncertainty. In addition, contact methods cannot be used for some delicate surfaces.

An attractive method for measuring a large radius of curvature is to utilize the resonant optical frequency of a Fabry-Pérot (FP) cavity [1,14,15]. In this method, the radius of curvature of a cavity mirror is calculated from the resonant mode spacing of the Gaussian (fundamental) mode and a high-order beam mode. The application scope of this method is basically concave mirror with high reflectivity. The finesse of the FP cavity, which is determined by the reflectivities of mirrors used in the FP cavity, should be enough high to separate the fundamental mode and a high-order beam mode. The advantages of this method are that it is non-contact, it has no moving parts, and it requires a relatively compact system. The measurement value in this method is a radio-frequency (RF) signal, which can be measured using a commercially available frequency counter. Generally, the standard of the frequency counter is an internal reference clock. The traceability of the reference clock to a time standard can be easily guaranteed and a relative accuracy of better than 10−9 can be realized. Therefore, this radius of curvature measurement is directly linked to the time standard [16] and reference standards of radii of curvature are not necessary. We have previously demonstrated radius of curvature measurements from 500 to 5000 mm with a repeatability of better than 3 × 10−5 using a sequential optical frequency locking procedure [15]. The proposed measurement system was realized using a low-finesse FP cavity, which consisted of flat and spherical mirrors, under usual air conditions. A relative spectroscopic uncertainty of better than 10−6 was achieved for a high-finesse FP cavity under vacuum conditions [14]. However, the actual uncertainty of the measured radius of curvature is limited by the surface figure of the flat mirror and is estimated to be on the order of 10−3 because an actual flat mirror has a locally finite radius of curvature. It is difficult to accurately measure the local value of the radius of curvature of a small area on a flat mirror illuminated by a laser beam. There is a possibility that unknown error due to the spherical component on the flat mirror is included in the measurement result.

In this paper, to overcome this problem, a three-spherical-mirror test is proposed. In the proposed method, three spherical mirrors are used and three FP cavities are sequentially constructed from each combination of two of the three spherical mirrors. Then, measurements of mode spacings for two spherical mirrors are obtained for each of the three FP cavities. The three unknown radii of curvature can be calculated by solving three simultaneous equations of the relations for each combination of two radii of curvature. The obtained radius of curvature can be used as the in situ reference value of a radius of curvature measurement system using the FP cavity. After the three-spherical-mirror test is completed, by replacing one spherical mirror in the FP cavity with another target spherical mirror, the radius of curvature of the target mirror can be accurately calculated using the reference value. Similar to the flatness calibration system using the Fizeau interferometer of which reference flat is calibrated by the three-flat-test, the radius of curvature measurement system, of which reference value is obtained by self-calibration method, can be constructed.

2. Measurement principle and problem in the flat mirror reference system

In an FP cavity consisting of two spherical mirrors separated by length Lab, where the mirrors have radii of curvature Ra and Rb, as shown in Fig. 1, a laser beam in Hermite-Gaussian mode resonates to the FP cavity if the optical frequency νm, j+k of the laser beam satisfies the following condition:

νm,j+k=c2nLab(m+(j+k+1)γ),
where
γ=1πcos1(1LabRa)(1LabRb),
and c is the speed of light in vacuum, n is the refractive index of air, m is the number of longitudinal (fundamental) modes, and j and k specify the orders of the transverse electromagnetic mode (TEMjk). If the spatial mode of the laser beam incident to the FP cavity is matched to that of the FP cavity, only the fundamental TEM00 modes appear. The free spectrum range (FSR) νFSRab of the FP cavity is given as

 figure: Fig. 1

Fig. 1 Basic configuration of the Fabry–Pérot cavity used in the proposed system.

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νFSRab=vm+1,0vm,0=c2nLab.

Figure 2 shows the transmission spectrum of the FP cavity with the beam profiles in the cross section. TEM00 is called the Gaussian mode, which is the fundamental mode, where νm,0 is the resonant frequency when the mode number is m. When the spatial mode of the incident laser beam is slightly mismatched to that of the FP cavity, transverse modes appear. The mode spacing νTRab between transverse modes is given as

νTRab=vm,1vm,0=1i(vm,ivm,0),
where νm,1 is the resonant frequency of TEM01 or TEM10 when the mode number is m, i = j + k, and νm,i is the resonant frequency of TEMjk when the mode number is m. From Eqs. (1), (3), and (4), γ is given as νTRab/νFSRab. In a previously proposed system [1,14,15], an FP cavity consisting of a flat mirror (Ra = ∞) and a spherical mirror (Rb = R) was used. The radius of curvature R in the FP cavity mirror can be calculated as
R=Lsin2(πνTRνFSR)=c2nνFSRsin2(πνTRνFSR).
The radius of curvature R in the FP cavity can be calculated from the FSR νFSR (cavity length L), the mode spacing νTR between transverse modes, and the refractive index of air n. Equation (5) is given under the assumption that the flat mirror is perfectly flat in the resonant area, which is the area illuminated by the laser beam. However, in practice, there is a spherical profile component even in the small resonant area on the flat mirror. The associated relative error due to the imperfection of the flat mirror is given as [14]
δRR=RLRe,
where δR is the error in the measured value of the radius of curvature R and Re is the value of the radius of curvature component in the resonant area of the flat mirror. If the diameter of the resonant area of the flat mirror is 0.4 mm, R = 10 m, L = 0.1 m, and δR/R = 10−3, then the sagitta of the curvature component in the resonant area of the flat mirror is only around 2 pm. It is difficult to accurately measure the value of the spherical profile component even in the small resonant area. Unknown error due to the spherical profile component is included in the radius of curvature measurement result using the flat mirror as the reference. The relative error becomes large as the radius of curvature becomes large.

 figure: Fig. 2

Fig. 2 Transmission spectrum lines of the FP cavity with transverse modes.

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3. Measurement principle of three-spherical-mirror test

To overcome the problem in the measurement system using a flat mirror as the ideal reference, a three-spherical-mirror test is proposed here. In the proposed method, three spherical mirrors with similar radii of curvature (Ra, Rb, and Rc) are used and three FP cavities are sequentially constructed from each combination of two of the three spherical mirrors (Ra and Rb, Ra and Rc, and Rb and Rc). Then, the FSR (cavity length) and the mode spacing between transverse modes are separately measured for each of the three FP cavities. As a result, the following three equations are obtained:

(1LabRa)(1LabRb)=cos(πvTRabvFSRab).
(1LacRa)(1LacRc)=cos(πvTRacvFSRac),
(1LbcRb)(1LbcRc)=cos(πvTRbcvFSRbc).
In Eqs. (7)–(9), there are three unknown radii of curvature (Ra, Rb, and Rc). The length Lab, Lac, and Lbc of each cavity can be experimentally obtained by Eq. (3). The mode spacings νTRab, νTRac, and νTRbc between transverse modes can be experimentally obtained by Eq. (4). In this work, three unknown values of radii of curvature in three simultaneous equations were numerically calculated using the Newton method. If the last combination in the three FP cavities gives Rb and Rc, the value of Rb can be used as the in situ reference value of the radius of curvature measurement system. The calculated value of radius of curvature Rb includes the deformation caused by the mirror holder and the additional local spherical profile component due to the surface forms on the spherical mirror. Theoretically, three unknown radii of curvature (Ra, Rb, and Rc) can be obtained by Eqs. (7)–(9) even if the three radii of curvature are not similar. However, readjustment to realize spatial mode matching is required in each of the three settings of FP cavity if three radii of curvature are not similar. It is better to prepare three spherical mirrors with similar radii of curvature in the practical point of view. After the three-spherical-mirror test is completed, by replacing one spherical mirror, whose radius of curvature is Rc, with another target spherical mirror, an accurate and wide-range radius of curvature measurement system, of which reference value is Rb, can be realized.

4. Experimental setup and measurement procedure

Figure 3 shows the optical setup of the proposed system. The system utilizes beat frequency measurement for the determination of FSRs, cavity lengths, and mode spacings between transverse modes. The basic configuration is almost the same as that used in [12]. Two commercially available external-cavity laser diodes (ECLDs) are used as frequency-tunable laser light sources (ECLD1, New Focus, Model 7000; ECLD2, Newport, Model 2010). The optical frequencies of the ECLDs can be tuned by more than 70 GHz at around 780 nm. The optical frequencies of the ECLDs are locked to the resonant frequencies of the FP cavity using electro-optic modulators (EOMs) [15]. The optical frequency f1 of ECLD1 and the optical frequency f2 of ECLD2 are modulated with frequencies of 40 and 5 MHz by EOM1 and EOM2 (New Focus, Model 4001), respectively. The two modulated beams are combined in a polarization-maintaining (PM) optical fiber and introduced into FP cavities with two of the three spherical mirrors. Spatial mode matching is realized using an adjustable fiber collimator (AFC). The beam diameter just after the AFC is around 2 mm. The intensity modulation signals generated around the resonance modes are detected by a photodetector. The error signals for locking the laser frequencies to the resonant frequencies are obtained from the intensity modulation signals through the respective band-pass filters and double-balanced mixers (DBMs). Optical frequencies f1 and f2 are locked to their respective resonant mode by the respective PI control loop. The optical beat frequency between f1 and f2 is measured by a frequency counter.

 figure: Fig. 3

Fig. 3 Experimental setup of the proposed system. ECLD: external-cavity laser diode, EOM: electro-optic modulator, AFC: adjustable fiber collimator, PBS: polarizing beam splitter, DBM: double-balanced mixer, PM: polarization-maintaining.

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The FP cavity was constructed on a low-thermal-expansion metal (Invar) base. Figure 4 shows a photograph of the FP cavity part. The two spherical mirrors are set using their respective holders, as shown in Fig. 4. In the three-spherical-mirror test, three FP cavities (Ra and Rb, Ra and Rc, and Rb and Rc) are sequentially constructed. For example, in the first FP cavity, spherical mirror Ra is set at the beam incident side and spherical mirror Rb is set at the beam output side. In the second FP cavity, spherical mirror Ra is continuously set at the beam incident side and spherical mirror Rc is set at the beam output side. In the third FP cavity, spherical mirror Rb is set at the beam incident side and spherical mirror Rc is continuously set at the beam output side. To realize the three combinations of FP cavities using the three spherical mirrors, one spherical mirror (Rb in this case) of the three is spatially flipped. In order to prevent any changes in the illumination area of the laser beam after a spherical mirror is flipped, the beam axis and the flip center are adjusted to coincide. The cavity length is around 102 mm and the FSR of the cavity is around 1.46 GHz. During the actual measurement, the FP cavity is simply covered to reduce air turbulence; there is no active control system for environmental conditions.

 figure: Fig. 4

Fig. 4 Photograph of the FP cavity part.

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As described in the previous section, the mode spacing values of νFSRab and νTRab, νFSRac and νTRac, and νFSRbc and νTRbc are required for the determination of the radii of curvature Ra, Rb, and Rc. In the proposed system, these values can be obtained from the beat frequency between optical frequencies f1 and f2. Ideally, optical frequencies f1 and f2 should be locked to the peaks of the resonant modes of the FP cavity by the EOM. However, in practice, the locked frequencies are slightly shifted from the peaks of the resonant frequencies by the residual amplitude modulation (RAM) caused by the EOM and other offset error sources in each locking system [17,18]. Although the axis of the EOM was adjusted at each setting to minimized the PAM, the difference between shift values, which are difficult to evaluate and be zero, causes error in the determination of the radius of curvature. To reduce this error, in the proposed system, optical frequency f1 is continuously locked to the resonant frequency νm-1,0 of the TEM00 mode with mode number m-1, and optical frequency f2 is sequentially locked to the resonant frequencies νm,0 of the TEM00 mode with mode number m, νm,i of the TEMjk (i = j + k) mode with mode number m, and νm+1,0 of the TEM00 mode with mode number m + 1. In this procedure, three beat frequencies, namely fbeat1, fbeat2, and fbeat3, are measured. They are given as

fbeat1=f2f1=vm,0vm1,0,fbeat2=vm,ivm1,0,fbeat3=vm+1,0vm1,0.
In our setup, fbeat1 is around 1.46 GHz and fbeat3 is around 2.93 GHz. When the FP cavity consists of Ra and Rb, mode spacing values of νFSRab and νTRab can be calculated as
νFSRab=fbeat3fbeat1=vm+1,0vm,0,νTRab=1i(fbeat2fbeat1)=1i(vm,ivm,0).
In these calculated values, common shift error values are canceled out because νm,0, νm,i, and νm+1,0 are realized by a common locking system. Therefore, Eq. (7) can produce accurate results from beat frequencies fbeat1, fbeat2, and fbeat3 and the refractive index of air n. The value of the refractive index of air can be calculated from environmental parameters using an empirical formula [19]. This measurement scheme is directly traceable to a time standard and the uncertainty of the frequency measurement by the frequency counter is negligibly small [15]. In addition, by increasing i, uncertainty due to beat frequency fluctuation in the determination of the mode spacing between transverse modes νTRab can be reduced to 1/i. Sets of mode spacing values, namely (νFSRac and νTRac) and (νFSRbc and νTRbc), can be similarly determined using sequential beat frequency measurements. Then, Eqs. (7)–(9) can be accurately determined and three simultaneous Eqs. (7)–(9) were numerically solved by using MATLAB (MathWorks, Inc.). The three unknown radii of curvature (Ra, Rb, and Rc) can be numerically calculated using the Newton method with high accuracy.

5. Experimental results

To demonstrate the performance of the three-spherical-mirror test, three spherical mirrors with radii of curvature of around 10 m and reflectivities of approximately 99.5% were used. Figure 5 shows an example of the actual error signals for the locking of the laser frequencies to the resonant frequencies of the FP cavity. In the first step of the alignment for each FP cavity, the optical system was adjusted so that only TEM00 appeared. Then, in the second step, the intensity of the transverse mode TEMjk was adjusted via the angle of the spherical mirror located at the back side of the FP cavity. In the experiments, the resonant frequency νm,3 (i.e., i = 3) was adopted to determine the mode spacing between transverse modes. Therefore, the influence of beat frequency fluctuations in the determination of νTRab, νTRac, and νTRbc was reduced to 1/3. As shown in Fig. 5, the intensity of νm,3 was set to be almost the same as that of νm,0 (TEM00 mode) in all measurements. The measured values of fbeat2 - fbeat1 were around 200 MHz and thus the values of νTRab, νTRac, and νTRbc were around 67 MHz.

 figure: Fig. 5

Fig. 5 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were both 10 m

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Figure 6 shows an example of the stability of beat frequency fbeat3. In all beat frequency measurements, the gate time of the frequency counter was 1 second and the signal-to-noise ratio (S/N) was better than 45 dB. From the measurement data shown in Fig. 6, the standard deviation of fbeat3 was approximately 0.68 kHz for a 2000-second measurement. Although the measurements were not conducted under vacuum conditions, as in [1,14], a sufficiently stable beat signal was achieved because optical frequencies f1 and f2 were locked to the same FP cavity. The relative measurement repeatabilities (standard deviations) of the FSR and the mode spacing between transverse modes were around 6.4 × 10−7 and 2.6 × 10−5, respectively.

 figure: Fig. 6

Fig. 6 Temporal variation in the measured beat frequency fbeat3.

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Table 1 shows the results of the three-spherical-mirror test. The test was repeated five times. The residual errors in Newton's method were negligible small in each three-spherical mirror test. The relative standard deviations of the values of Ra, Rb, and Rc were around 1.4 × 10−4, 2.0 × 10−4, and 1.0 × 10−4, respectively. Compared with the relative measurement repeatabilities of the FSR and the mode spacing between transverse modes, the relative repeatability of the three-spherical-mirror test is large. The repeatability of the three-spherical-mirror test mainly depends on the repeatability of the alignment in the second step described above. The angle of the spherical mirror located at the back side of the FP cavity was adjusted in each measurement. This adjustment changed the resonant area, which in turn affected the effective value of the radius of curvature because of the imperfections of the surface forms of the spherical mirrors.

Tables Icon

Table 1. Measurement results of three-spherical-mirror test

It is estimated that there are four dominant error sources for absolute accuracy of measured radii of curvature: beat frequency measurement error, frequency shift caused by the residual amplitude modulation (RAM) caused by the EOM and other offset error sources in each locking system, repeatability of the mode spacing measurement, and repeatability of three-spherical-mirror test itself.

The measurement values fbeat1, fbeat2, and fbeat3 are RF signals, which can be measured by a commercially available frequency counter. Generally, the standard of a frequency counter is an internal reference clock. The traceability of the reference clock to a time standard can be easily guaranteed. An accuracy of better than 10−9 can be realized. Therefore, the proposed radius of curvature measurement is directly linked to the time standard; the uncertainties from the beat frequency measurement are negligibly small. The error value due to the frequency shift can be estimated by the equality of the mode spacings νTR [15]. The equality was sufficiently smaller than the repeatability of mode spacing measurement. The relative repeatability of mode spacing measurements was better than 1 × 10−5. As described above, the repatablity of the three-spherical-mirror test was estimated to be 1.5 × 10−4. From these error evaluations, the uncertainty of the proposed method is mostly determined by the repeatability of the three-spherical-mirror test, and was estimated to be around 1.5 × 10−4.

Ideally, it is recommended that the number of measurements is 10 or more when the uncertainty is evaluated from the measurement repeatability. However, three-spherical-mirror test requires a long waiting time to stabilize the measurement system. Multiple conduction of three-spherical-mirror test may lead to other drift errors. Therefore, the uncertainty was estimated from results of five times test. In the theory of the uncertainty analysis, standard uncertainty estimated from five data is 1.08 times larger than that estimated from ten data [20].

One spherical mirror used in the last combination in the three FP cavities can be used as the reference spherical mirror for the radius of curvature measurement system. The value of the radius of curvature was determined from the three-spherical-mirror test in situ. By replacing the other spherical mirror located at the back side of the FP cavity with another target spherical mirror and measuring fbeat1, fbeat2, and fbeat3, the radius of curvature of the target spherical mirror can be determined. To demonstrate the capability of measuring the radius of curvature of a spherical mirror, after the three-spherical-mirror test was completed, the spherical mirror located at the back side of the FP cavity in the last combination was replaced with a target spherical mirror whose radius of curvature Rd was around 20 m. When the Rc is used as the reference value, the value of the radius of curvature Rd is given as

Rd=Lcd1cos2(πνTRcdνFSRcd)1LcdRc.

Figure 7 shows the error signals for the locking of the laser frequencies to the resonant frequencies of the FP cavity for spherical mirrors with radii of curvature of 10 and 20 m, respectively. Signals similar to those in Fig. 5 were obtained. The measured values of fbeat2 - fbeat1 were around 172 MHz and thus the mode spacing νTRcd between transverse modes was around 57 MHz. The measured radius of curvature of the target spherical mirror was 20.338 m and the relative measurement repeatability was around 7.2 × 10−5. The measurement uncertainty was estimated from the measurement repeatabilities of νTRcd,νFSRcd, and Lcd, and an estimated uncertainty of the reference spherical mirror, whose radius of curvature was around 10 m as determined from the three-spherical-mirror test, of 1.5 × 10−4. The value of the standard uncertainty was calculated by using an Monte Carlo simulation and relative standard uncertainty was calculated to be around 3.1 × 10−4. The dominant uncertainty source was the uncertainty of the reference spherical mirror of 1.5 × 10−4, because measurement repeatabilities of beat frequencies were enough small.

 figure: Fig. 7

Fig. 7 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were 10 and 20 m, respectively.

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6. Conclusion

An accurate measurement system, especially for radii of curvature larger than 10 m level, that uses a three-spherical-mirror test was developed. By using the three-spherical-mirror test in an FP cavity, the local value of the radius of curvature of a mirror can be determined in situ and this mirror can then be used as the reference spherical mirror in a radius of curvature measurement system. We demonstrated determinations of radii of curvature of around 10 m using the three-spherical-mirror test with uncertainties of around 1.5 × 10−4 and then demonstrated measurement of radius of curvature of around 20 m with uncertainties of around 3.1 × 10−4 by using the spherical mirror, of which radius of curvature was determined by the three-spherical-mirror test, as the reference sphere. The proposed system has high practical applicability because measurements can be conducted under usual air conditions with only slight changes to settings and the measurement results are directly traceable to the time standard because beat frequency measurement is used. The proposed method is not applicable for spherical surfaces with low reflectivity. However, a spherical mirror measured by the proposed system can be used as the reference standard in practical optical systems. The measured radius of curvature corresponds to the average radius of curvature over the area illuminated by the laser beam, which is smaller than approximately 1 mm2 in the present system. Therefore, as in the interferometric method, to obtain the average radius of curvature over a large area, it is necessary to move the target spherical mirror and measure it in several positions.

Funding

Precise Measurement Technology Promotion Foundation (PMTP-F); Mitutoyo Association for Science and Technology (MAST).

References

1. N. Uehara and K. Ueda, “Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry-Perot interferometer,” Appl. Opt. 34(25), 5611–5619 (1995). [CrossRef]   [PubMed]  

2. Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 808–825.

3. M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

4. W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013). [CrossRef]  

5. B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).

6. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal radius measurement,” Opt. Express 18(3), 2345–2360 (2010). [CrossRef]   [PubMed]  

7. J. Yang, L. Qiu, W. Zhao, X. Zhang, and X. Wang, “Radius measurement by laser confocal technology,” Appl. Opt. 53(13), 2860–2865 (2014). [CrossRef]   [PubMed]  

8. M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980). [CrossRef]  

9. W. Zhao, X. Zhang, Y. Wang, and L. Qiu, “Laser reflection differential confocal large-radius measurement,” Appl. Opt. 54(31), 9308–9314 (2015). [CrossRef]   [PubMed]  

10. Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001). [CrossRef]   [PubMed]  

11. Q. Wang, U. Griesmann, and J. A. Soons, “Holographic radius test plates for spherical surfaces with large radius of curvature,” Appl. Opt. 53(20), 4532–4538 (2014). [CrossRef]   [PubMed]  

12. H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005). [CrossRef]  

13. C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005). [CrossRef]  

14. J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009). [CrossRef]  

15. Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018). [CrossRef]  

16. Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009). [CrossRef]   [PubMed]  

17. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985). [CrossRef]  

18. Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016). [CrossRef]   [PubMed]  

19. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35(9), 1566–1573 (1996). [CrossRef]   [PubMed]  

20. JCGM 2008, Evaluation of Measurement Data - Guide to the Expression of Uncertainty in Measurement, (BIPM, 2008).

References

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  1. N. Uehara and K. Ueda, “Accurate measurement of the radius of curvature of a concave mirror and the power dependence in a high-finesse Fabry-Perot interferometer,” Appl. Opt. 34(25), 5611–5619 (1995).
    [Crossref] [PubMed]
  2. Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 808–825.
  3. M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).
  4. W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
    [Crossref]
  5. B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).
  6. W. Zhao, R. Sun, L. Qiu, and D. Sha, “Laser differential confocal radius measurement,” Opt. Express 18(3), 2345–2360 (2010).
    [Crossref] [PubMed]
  7. J. Yang, L. Qiu, W. Zhao, X. Zhang, and X. Wang, “Radius measurement by laser confocal technology,” Appl. Opt. 53(13), 2860–2865 (2014).
    [Crossref] [PubMed]
  8. M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
    [Crossref]
  9. W. Zhao, X. Zhang, Y. Wang, and L. Qiu, “Laser reflection differential confocal large-radius measurement,” Appl. Opt. 54(31), 9308–9314 (2015).
    [Crossref] [PubMed]
  10. Y. Xiang, “Focus retrocollimated interferometry for long-radius-of-curvature measurement,” Appl. Opt. 40(34), 6210–6214 (2001).
    [Crossref] [PubMed]
  11. Q. Wang, U. Griesmann, and J. A. Soons, “Holographic radius test plates for spherical surfaces with large radius of curvature,” Appl. Opt. 53(20), 4532–4538 (2014).
    [Crossref] [PubMed]
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  13. C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
    [Crossref]
  14. J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
    [Crossref]
  15. Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
    [Crossref]
  16. Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
    [Crossref] [PubMed]
  17. N. C. Wong and J. L. Hall, “Servo control of amplitude modulation in frequency-modulation spectroscopy: demonstration of shot-noise-limited detection,” J. Opt. Soc. Am. B 2(9), 1527–1533 (1985).
    [Crossref]
  18. Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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2018 (1)

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

2016 (1)

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (3)

2013 (1)

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

2010 (1)

2009 (2)

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
[Crossref] [PubMed]

2005 (2)

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

2001 (1)

1996 (1)

1995 (1)

1985 (1)

1983 (1)

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

1980 (1)

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Bitou, Y.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Y. Bitou, “Displacement metrology directly linked to a time standard using an optical-frequency-comb generator,” Opt. Lett. 34(10), 1540–1542 (2009).
[Crossref] [PubMed]

Burge, J. H.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Ciddor, P. E.

Gao, D.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Gerchman, M. C.

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Griesmann, U.

Guo, J.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Hall, J. L.

Hunter, G. C.

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

Lawall, J. R.

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Meng, J.

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Murtyand, M. V. R. K.

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

Pratt, J. R.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Qiu, L.

Sha, D.

Shukla, R. P.

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

Soons, J. A.

Sun, R.

Takei, Y.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Takeuchi, H.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Telada, S.

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Truax, B.

B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).

Tsutsumi, H.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Ueda, K.

Uehara, N.

Wang, Q.

Wang, X.

Wang, Y.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

W. Zhao, X. Zhang, Y. Wang, and L. Qiu, “Laser reflection differential confocal large-radius measurement,” Appl. Opt. 54(31), 9308–9314 (2015).
[Crossref] [PubMed]

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Wong, N. C.

Xiang, Y.

Yang, J.

Yoshizumi, K.

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Yu, Y.

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Zehnder, R.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Zhang, X.

Zhao, C.

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Zhao, W.

Appl. Opt. (6)

J. Opt. Soc. Am. B (1)

Laser Focus World (1)

B. Truax, “Interferometry: Achieving precision radius metrology for large optics,” Laser Focus World 50(4), 65–69 (2014).

Meas. Sci. Technol. (1)

J. R. Lawall, “High resolution determination of radii of curvature using Fabry-Perot interferometry,” Meas. Sci. Technol. 20(4), 045302 (2009).
[Crossref]

Opt. Commun. (1)

W. Zhao, J. Guo, L. Qiu, Y. Wang, J. Meng, and D. Gao, “Low transmittance ICF capsule geometric parameters measurement using laser differential confocal technique,” Opt. Commun. 292, 62–67 (2013).
[Crossref]

Opt. Eng. (3)

M. V. R. K. Murtyand and R. P. Shukla, “Measurement of long radius of curvature,” Opt. Eng. 22(2), 222231 (1983).

M. C. Gerchman and G. C. Hunter, “Differential technique for accurately measuring the radius of curvature of long radius concave optical surfaces,” Opt. Eng. 19(6), 843–848 (1980).
[Crossref]

C. Zhao, R. Zehnder, and J. H. Burge, “Measuring the radius of curvature of a spherical mirror with an interferometer and a laser tracker,” Opt. Eng. 44(9), 090506 (2005).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Precis. Eng. (1)

Y. Bitou, Y. Takei, and S. Telada, “Accurate and wide-range radius of curvature measurement directly linked to a time standard using a Fabry–Pérot cavity,” Precis. Eng. 54, 149–153 (2018).
[Crossref]

Proc. SPIE (1)

H. Tsutsumi, K. Yoshizumi, and H. Takeuchi, “Ultrahighly accurate 3D profilometer,” Proc. SPIE 5638, 387–394 (2005).
[Crossref]

Rev. Sci. Instrum. (1)

Y. Yu, Y. Wang, and J. R. Pratt, “Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer,” Rev. Sci. Instrum. 87(3), 033101 (2016).
[Crossref] [PubMed]

Other (2)

JCGM 2008, Evaluation of Measurement Data - Guide to the Expression of Uncertainty in Measurement, (BIPM, 2008).

Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), pp. 808–825.

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Figures (7)

Fig. 1
Fig. 1 Basic configuration of the Fabry–Pérot cavity used in the proposed system.
Fig. 2
Fig. 2 Transmission spectrum lines of the FP cavity with transverse modes.
Fig. 3
Fig. 3 Experimental setup of the proposed system. ECLD: external-cavity laser diode, EOM: electro-optic modulator, AFC: adjustable fiber collimator, PBS: polarizing beam splitter, DBM: double-balanced mixer, PM: polarization-maintaining.
Fig. 4
Fig. 4 Photograph of the FP cavity part.
Fig. 5
Fig. 5 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were both 10 m
Fig. 6
Fig. 6 Temporal variation in the measured beat frequency fbeat3.
Fig. 7
Fig. 7 Error signals for locking the laser frequency to the resonant frequencies of the FP cavity. The radii of curvature of the two spherical mirrors were 10 and 20 m, respectively.

Tables (1)

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Table 1 Measurement results of three-spherical-mirror test

Equations (12)

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ν m,j+k = c 2n L ab ( m+(j+k+1)γ ),
γ= 1 π cos 1 ( 1 L ab R a )( 1 L ab R b ) ,
ν FSRab = v m+1,0 v m,0 = c 2nL ab .
ν TRab = v m,1 v m,0 = 1 i ( v m,i v m,0 ),
R= L sin 2 ( π ν TR ν FSR ) = c 2n ν FSR sin 2 ( π ν TR ν FSR ) .
δR R = RL R e ,
( 1 L ab R a )( 1 L ab R b ) =cos( π v TRab v FSRab ).
( 1 L ac R a )( 1 L ac R c ) =cos( π v TRac v FSRac ),
( 1 L bc R b )( 1 L bc R c ) =cos( π v TRbc v FSRbc ).
f beat1 = f 2 f 1 = v m,0 v m1,0 , f beat2 = v m,i v m1,0 , f beat3 = v m+1,0 v m1,0 .
ν FSRab = f beat3 f beat1 = v m+1,0 v m,0 , ν TRab = 1 i ( f beat2 f beat1 )= 1 i ( v m,i v m,0 ).
R d = L cd 1 cos 2 ( π ν TRcd ν FSRcd ) 1 L cd R c .

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