Balanced plane-mirror heterodyne interferometer with subnanometer periodic nonlinearity Peng-cheng Hu,* Peng Chen, Xue-mei Ding, and Jiu-bin Tan Institute of Ultra-precision Optoelectronic Instrument Engineering, Harbin Institute of Technology, Harbin 150080, China *Corresponding author: [email protected] Received 9 May 2014; revised 18 July 2014; accepted 18 July 2014; posted 21 July 2014 (Doc. ID 211564); published 15 August 2014
A balanced plane-mirror heterodyne interferometer with a polarizing beam splitter used to recombine the reference and measurement beams is proposed to reduce periodic nonlinearity and to eliminate thermal error. Experimental results indicated that the periodic error due to ghost reflection was kept within 36 pm, and the interferometer proposed was immune from thermal error. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.3930) Metrological instrumentation; (120.5050) Phase measurement; (120.6810) Thermal effects. http://dx.doi.org/10.1364/AO.53.005448
1. Introduction
Heterodyne interferometers are widely used for length measurement in precision engineering, metrology, lithography applications, and advanced scientific applications [1] because of their wide dynamic range, high signal-to-noise ratio, almost unlimited resolution, and direct traceability to the length standards [2]. However, the resolution provided by a heterodyne interferometer is much smaller than subnanometer; the accuracy achievable with a heterodyne interferometer is limited by periodic nonlinearity [3]. Much work has been done on the theoretical models of periodic nonlinearity [1,4–6]. Previous research reveals that periodic nonlinearity originates from a mixed heterodyne source [5] and nonperfect polarizing optics [4]. So when a two-frequency orthogonally polarized source is used in a traditional heterodyne interferometer, there will be nonorthogonal, slightly elliptical beams to contribute periodic nonlinearity errors. These combinations with imperfect polarizing optics, polarization alignment between the source and optics, and ghost reflections create additional periodic nonlinearity errors [4,6,7]. Periodic nonlinearity could be reduced using either compensation techniques 1559-128X/14/245448-05$15.00/0 © 2014 Optical Society of America 5448
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[1,8–10] or two spatially separated beams [10,11]. For example, Joo et al. recently proposed three novel heterodyne interferometer configurations without periodic nonlinearity [12,13]. A retroreflector is used in three interferometers to recombine the crossed reference and measurement beams. The drawback of a plane-mirror interferometer [13] is that the reference and measurement paths are not perfectly balanced. These slightly unbalanced paths, especially those in linear plane-mirror interferometers, cause significant thermal error from temperature fluctuations during normal laboratory tests. Therefore, a balanced plane-mirror heterodyne interferometer with a polarizing beam splitter (PBS) used to recombine reference and measurement beams is proposed to reduce periodic nonlinearity and to eliminate thermal error. 2. Balanced Plane-Mirror Heterodyne Interferometer
As shown in Fig. 1, two beams propagate to a beam splitter (BS), where they are split equally. Reflected beams transmit in the BS and are reflected by mirror 1 (M1), and then travel back to PBS2 where the two beams are recombined to create a beat signal with frequency f s f s f 2 − f 1 detected by the reference photodetector (PDR ). The measurement and reference beams are the horizontally polarized beam with optical frequency
Fig. 1. Schematic diagram of balanced plane-mirror heterodyne interferometer. BS, beams splitter; PBS1 and PBS2, polarizing beam splitters; QWP1 and QWP2, quarter-wave plates; RAP, right angle prism; M1, M2, and M3, mirrors; AP1 and AP2, absorber plates; PDR and PDM , reference and measurement photodetectors.
f 1 and the vertically polarized beam with optical frequency f 2, respectively. The measurement beam passes through PBS1 and quarter-wave plate 1 (QWP1) and travels toward M2. After being reflected from M2, the measurement beam passes through QWP1 and becomes vertically polarized. The measurement beam is then reflected by PBS1 to a right angle prism (RAP) and is horizontally displaced by the RAP. The measurement beam is reflected by PBS1 once again and then passes through QWP1 for the third time, is reflected from M2 for the second time, and passes through QWP1 for the last time. The measurement beams is then horizontally polarized once again and passes through PBS1 into PBS2. The reference beam is reflected from PBS1, passes through QWP2, and travels toward M3. After being reflected from M3, the reference beam passes through QWP2 and becomes horizontally polarized. The polarized reference beam transmits through PBS1 to RAP and is horizontally displaced by RAP. The reference beam transmits through PBS1 once again and passes through QWP2 for the third time, is reflected by M3 for the second time, and passes through QWP2 for the last time. The reference beam is then vertically polarized once again and is reflected by PBS1 into PBS2. The measurement and reference beams are recombined in PBS2 to create a beat signal detected by the measurement PD (PDM ). In addition, two absorber plates (AP1 and AP2) are used to absorb the leakage of the measurement and reference beams from PBS1. The main advantage of the proposed interferometer is that there is no beam leakage from the optics, which means periodic nonlinearity is essentially eliminated. Second, the optical paths in the optics of the reference and measurement beams are balanced by optimizing optical components, which eliminates thermal errors and common-mode errors. Third, the laser beam can be delivered by polarization maintaining (PM) fiber to relax the requirement for optical alignment and keep possible heat sources away from the system.
As shown in Fig. 2, in order to eliminate polarization mixing or frequency mixing of a traditional heterodyne laser source, feasibility experiments were run using an optical source with two spatially separated beams driven by two acousto-optic frequency shifters (AOFSs) [13]. Two AOFSs (AOFS1, AOFS2, MT80-B30A1-VIS, A.A.) were employed to generate two acousto-optic frequencies of δf 1 and δf 2 from a stabilized single frequency (f 0 ) source (λ0 632.8 nm, 25-STP-910-230, Melles Griot). The first beam from AOFS1 had frequency f 1 f 0 δf 1 , and the second beam from AOFS2 had frequency f 2 f 0 δf 2 . So beat frequency f s f s f 2 − f 1 f 0 δf 2 − f 0 δf 1 δf 2 − δf 1 was created, and none of the beams had a leakage component to cause periodic nonlinearity. The beams with f 0 and other beams with higher orders (except first order) from AOFS1 and AOFS2 were blocked with pinholes (PHs), which were approximately 200 mm apart from the AOFSs. The two beams from AOFSs were adjusted and aligned before they entered the interferometer. The polarization direction was adjusted by two half-wave plates (HWP1 and HWP2). In an ideal case, the irradiances at PDR and PDM can be expressed as I r ∝ cos2πf 1 t θ1 cos2πf 2 t θ2 ;
(1)
I m ∝ cos2πf 1 t θ1 δ1 θm cos2πf 2 t θ2 δ2 ; (2) where θ1 and θ2 are the phase offsets for AOFS1 and AOFS2, δ1 and δ2 are the phase offsets in the optics paths for measurement and reference beams, and θm is the Doppler phase driven by the measurement mirror. By substituting f s f 2 − f 1 and ignoring the optical frequencies too high to detect, the measured irradiances can be simplified as shown: I r ∝ cos2πf s t θ1 − θ2 ;
(3)
I m ∝ cos2πf s t θ1 –θ2 δ1 –δ2 θm :
(4)
The common-mode errors and thermal errors in the interferometer can be eliminated by designing balanced optics paths for the measurement and reference beams, so that δ1 –δ2 0. The measured
Fig. 2. Optical source with two spatially separated beams driven by two AOFSs at slightly different frequencies from a stabilized source. BS, beam splitter; AOFS1 and AOFS2, acousto-optic frequency shifters; HWP1 and WHP2, half-wave plates; PH, pinhole. 20 August 2014 / Vol. 53, No. 24 / APPLIED OPTICS
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phase difference between I r and I m can be expressed as ϕmr ϕm − ϕr θm . Clearly, the measuring phase after the AOFSs cancels the AOFSs’ differential phase shift. If two spatially separated beams coming out from the optical source are delivered by two PM fibers, the fiber-induced drifts would be included in θ1 and θ2 and canceled in the measuring phase as well. 3. Results and Discussion
As described by Badami and Patterson [14], the actual (nonideal) time-varying component of the PDM signal in a traditional heterodyne interferometer can be expressed as I m ∝ jA1 j2 jA2 j2 jε12 j2 jε21 j2 |{z} (
DCterms
2R A1 A2 ei2πf s tϕm |{z} Base signal
A2 ε21
A1 ε12 ei2πf s t |{z } First harmonic error terms
)
ε12 ε21 ei2πf s t−ϕm |{z} Second harmonic error term
A1 ε21 A2 ε12 e−iϕm ; |{z}
(5)
Quasi-DCterms
where A1 and A2 are the amplitudes of the measurement and reference beams, respectively, and ε12 and ε21 are the deviations from ideal measurement inputs. So the first choice was to directly measure the magnitudes of first and second harmonic errors using the frequency spectrum of output PDM (the so-called frequency domain method) [14,15]. With δf 1 80.00 MHz, δf 2 80.50 MHz, f s 500 kHz, target M2 was moved at a velocity of approximately 11.12 mm/s to yield Doppler shift Δf d 70.30 kHz. The PDM signal was received and then input into a spectrum analyzer (Agilent MSO9254A). As shown in Fig. 3, the original beat frequency is shifted from
500 to 570.30 kHz. There are no recognizable nanoscale periodic errors with noise level ≤ 0.4 nm at first harmonic frequency (500 kHz) or second harmonic frequency (429.70 kHz), which means, compared to several nanometers in a traditional interferometer, the periodic error has been reduced by the proposed interferometer to a subnanometer level. The phase quadrature measurement method was then used to measure the residual periodic errors. As reported by Wu et al. in [11], the relation between periodic phase error (dϕ) and amplitude change (dR∕R) can be expressed as dR : (6) jdϕj R With δf 1 80.00 MHz, δf 2 80.06 MHz, f s 60 kHz, target M2 was moved at 1.4 μm/s to yield Doppler shift Δf d 9 Hz. The PDR signal was received as a reference input, and the PDM signal was received and then demodulated in a digital lock-in amplifier (SR830). The upper limit of the digital lock-in amplifier was 102 kHz (60 kHz < 102 kHz, second harmonic frequency of the detectors 120 kHz > 102 kHz), and so f s 60 kHz was a right choice. The output quadrature components of the digital lock-in amplifier were acquired and offline calculated using MATLAB software. The calculation with MATLAB could be expressed as 8 Rn C2 n S2 n1∕2 ; > > > > ϕn tan−1 Sn∕Cn; > > Rn− ; dϕn dRn > ¯ ¯ R R > > > Filterdϕn; > : λ0 dϕn; dLn 14 2π
and FTdL
X Δn i1
FFTdLi:Δn:N · window
2 ; N (8)
Fig. 3. Periodic errors measured using frequency domain method at first harmonic frequency (500 kHz) and second harmonic frequency (429.70 kHz). 5450
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where n is the index of sample data; N is the number of samples; 4 is number of optical paths to target M2; Cn, Sn are the output quadrature components (cosine, sine) of the digital lock-in amplifier; and dLn is the displacement corresponding to phase dϕn. FFTdLi:Δn:N · window includes the data sampled for the second time at Δn intervals for an appropriate frequency range, the addition of a window function to reduce the fast Fourier transform (FFT) errors, and the calculation of FFT. The purpose of Eq. (8) is to reduce the white noise by doing Δn loops and averaging. FTdL is the result of the Fourier transform of dLn. As shown in Fig. 4, the first harmonic peak amplitude (at 9 Hz) is 36 pm, and the second harmonic peak amplitude (at 18 Hz) is 10 pm. No imperfect optics can cause a periodic error in the proposed interferometer, and the electronic demodulation periodic error of the digital
Fig. 5. Experimental result of thermal error. The error curve (dotted) represents the case with a glass plate 3 mm thick of the same material as PBS1 placed closely between QWP1 and M2; the error curve (solid) represents the other case without the glass plate.
error compared to the unbalanced case. An ideal PBS1 can cancel the slight influence of thermal error while the proposed interferometer is used for displacement measurement. 4. Conclusion
Fig. 4. Periodic errors with 1.4 μm/s from phase quadrature measurement method. (a) Demodulated periodic phase measured with a digital lock-in amplifier; (b) periodic error calculated with 1∕4 λ0 ∕2π dR∕R (4 is number of optical paths to target M2, λ0 632.8 nm is laser wavelength); (c) FFT result of periodic error 4(b). The dominant frequency is approximately 9 Hz, the first harmonic peak amplitude (at 9 Hz) is 36 pm, and the second harmonic peak amplitude (at 18 Hz) is 10 pm.
lock-in amplifier is less than 2.5 pm [11]. The second harmonic frequency of the detectors is out of the range of the digital lock-in amplifier. Shot noise of the detectors and white noise of data acquisition circuit are not periodic and can be eliminated by filtering and averaging. The variation of laser amplitude has nothing to do with the first and second order optical period, and their influence can be minimized by averaging. The periodic stage motion errors and mount vibration errors have nothing to do with the first and second order optical period by choosing the target moving speed. The FFT errors can be reduced by using a window function. Thus, the peak of the periodic error in Fig. 4(c) is attributed to ghost reflection [7]. The immunity of the proposed interferometer from thermal error was investigated with a temperature increment of 1°C per hour from 20°C to 26°C. As shown in Fig. 5, the thermal error achieved with a 3 mm glass plate placed closely between QWP1 and M2 has an approximately linear growth as temperature rises. The thermal error achieved without a 3 mm glass plate goes down to 2 nm, which means the proposed interferometer is immune from thermal
A balanced plane-mirror heterodyne interferometer with a PBS used to recombine the reference and measurement beams was proposed to reduce periodic nonlinearity and to eliminate thermal error for displacement measurement. Feasibility experiments were run using an optical source with two spatially separated beams driven by two AOFSs to eliminate polarization mixing or frequency mixing. Experimental results indicated that the periodic error due to ghost reflection was kept within 36 pm, and the interferometer proposed was immune from thermal error. In practice, the laser beam of the interferometer can be delivered by PM fiber to relax the requirement for optical alignment and keep possible heat sources away from the system. This work was funded by National Natural Science Foundation of China (Nos. 51105114, 51205091 and 51305105). References 1. C. M. Wu and R. D. Deslattes, “Analytical modeling of the periodic nonlinearity in heterodyne interferometry,” Appl. Opt. 37, 6696–6700 (1998). 2. J. Ahn, J. A. Kim, C. S. Kang, J. W. Kim, and S. Kim, “High resolution interferometer with multiple pass optical configuration,” Opt. Express 17, 21042–21049 (2009). 3. T. Eom, J. Y. Kim, and K. Jeong, “The dynamic compensation of nonlinearity in a homodyne laser interferometer,” Meas. Sci. Technol. 12, 1734–1738 (2001). 4. S. J. A. G. Cosijns, H. Haitjema, and P. H. J. Schellekens, “Modeling and verifying non-linearities in heterodyne displacement interferometry,” Precis. Eng. 26, 448–455 (2002). 5. A. E. Rosenbluth and N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990). 6. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30, 337–346 (2006). 7. C. M. Wu, “Periodic nonlinearity resulting from ghost reflections in heterodyne interferometry,” Opt. Commun. 215, 17–23 (2003). 8. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992). 20 August 2014 / Vol. 53, No. 24 / APPLIED OPTICS
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9. H. Haitjema, S. J. A. G. Cosijns, N. J. J. Roset, and M. J. Jansen, “Improving a commercially available heterodyne laser interferometer to sub-nm uncertainty,” Proc. SPIE 5190, 347–354 (2003). 10. C. Weichert, P. Köchert, R. Köning, J. Flügge, B. Andreas, U. Kuetgens, and A. Yacoot, “A heterodyne interferometer with periodic nonlinearities smaller than ±10 pm,” Meas. Sci. Technol. 23, 094005 (2012). 11. C. M. Wu, J. Lawall, and R. D. Deslattes, “Heterodyne interferometer with subatomic periodic nonlinearity,” Appl. Opt. 38, 4089–4094 (1999). 12. K. Joo, J. D. Ellis, J. W. Spronck, P. J. M. v. Kan, and R. H. M. Schmidt, “Simple heterodyne laser interferometer with
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subnanometer periodic errors,” Opt. Lett. 34, 386–388 (2009). 13. K. Joo, J. D. Ellis, E. S. Buice, J. W. Spronck, and R. H. M. Schmidt, “High resolution heterodyne interferometer without detectable periodic nonlinearity,” Opt. Express 18, 1159–1165 (2010). 14. V. G. Badami and S. R. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24, 41–49 (2000). 15. T. L. Schmitz and J. F. Beckwith, “Acousto-optic displacementmeasuring interferometer: a new heterodyne interferometer with Angstrom-level periodic error,” J. Mod. Opt. 49, 2105– 2114 (2002).
A balanced plane-mirror heterodyne interferometer with a polarizing beam splitter used to recombine the reference and measurement beams is proposed to reduce periodic nonlinearity and to eliminate thermal error. Experimental results indicated that the periodic error due to ghost reflection was kept within 36 pm, and the interferometer proposed was immune from thermal error. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (120.3180) Interferometry; (120.3930) Metrological instrumentation; (120.5050) Phase measurement; (120.6810) Thermal effects. http://dx.doi.org/10.1364/AO.53.005448
1. Introduction
Heterodyne interferometers are widely used for length measurement in precision engineering, metrology, lithography applications, and advanced scientific applications [1] because of their wide dynamic range, high signal-to-noise ratio, almost unlimited resolution, and direct traceability to the length standards [2]. However, the resolution provided by a heterodyne interferometer is much smaller than subnanometer; the accuracy achievable with a heterodyne interferometer is limited by periodic nonlinearity [3]. Much work has been done on the theoretical models of periodic nonlinearity [1,4–6]. Previous research reveals that periodic nonlinearity originates from a mixed heterodyne source [5] and nonperfect polarizing optics [4]. So when a two-frequency orthogonally polarized source is used in a traditional heterodyne interferometer, there will be nonorthogonal, slightly elliptical beams to contribute periodic nonlinearity errors. These combinations with imperfect polarizing optics, polarization alignment between the source and optics, and ghost reflections create additional periodic nonlinearity errors [4,6,7]. Periodic nonlinearity could be reduced using either compensation techniques 1559-128X/14/245448-05$15.00/0 © 2014 Optical Society of America 5448
APPLIED OPTICS / Vol. 53, No. 24 / 20 August 2014
[1,8–10] or two spatially separated beams [10,11]. For example, Joo et al. recently proposed three novel heterodyne interferometer configurations without periodic nonlinearity [12,13]. A retroreflector is used in three interferometers to recombine the crossed reference and measurement beams. The drawback of a plane-mirror interferometer [13] is that the reference and measurement paths are not perfectly balanced. These slightly unbalanced paths, especially those in linear plane-mirror interferometers, cause significant thermal error from temperature fluctuations during normal laboratory tests. Therefore, a balanced plane-mirror heterodyne interferometer with a polarizing beam splitter (PBS) used to recombine reference and measurement beams is proposed to reduce periodic nonlinearity and to eliminate thermal error. 2. Balanced Plane-Mirror Heterodyne Interferometer
As shown in Fig. 1, two beams propagate to a beam splitter (BS), where they are split equally. Reflected beams transmit in the BS and are reflected by mirror 1 (M1), and then travel back to PBS2 where the two beams are recombined to create a beat signal with frequency f s f s f 2 − f 1 detected by the reference photodetector (PDR ). The measurement and reference beams are the horizontally polarized beam with optical frequency
Fig. 1. Schematic diagram of balanced plane-mirror heterodyne interferometer. BS, beams splitter; PBS1 and PBS2, polarizing beam splitters; QWP1 and QWP2, quarter-wave plates; RAP, right angle prism; M1, M2, and M3, mirrors; AP1 and AP2, absorber plates; PDR and PDM , reference and measurement photodetectors.
f 1 and the vertically polarized beam with optical frequency f 2, respectively. The measurement beam passes through PBS1 and quarter-wave plate 1 (QWP1) and travels toward M2. After being reflected from M2, the measurement beam passes through QWP1 and becomes vertically polarized. The measurement beam is then reflected by PBS1 to a right angle prism (RAP) and is horizontally displaced by the RAP. The measurement beam is reflected by PBS1 once again and then passes through QWP1 for the third time, is reflected from M2 for the second time, and passes through QWP1 for the last time. The measurement beams is then horizontally polarized once again and passes through PBS1 into PBS2. The reference beam is reflected from PBS1, passes through QWP2, and travels toward M3. After being reflected from M3, the reference beam passes through QWP2 and becomes horizontally polarized. The polarized reference beam transmits through PBS1 to RAP and is horizontally displaced by RAP. The reference beam transmits through PBS1 once again and passes through QWP2 for the third time, is reflected by M3 for the second time, and passes through QWP2 for the last time. The reference beam is then vertically polarized once again and is reflected by PBS1 into PBS2. The measurement and reference beams are recombined in PBS2 to create a beat signal detected by the measurement PD (PDM ). In addition, two absorber plates (AP1 and AP2) are used to absorb the leakage of the measurement and reference beams from PBS1. The main advantage of the proposed interferometer is that there is no beam leakage from the optics, which means periodic nonlinearity is essentially eliminated. Second, the optical paths in the optics of the reference and measurement beams are balanced by optimizing optical components, which eliminates thermal errors and common-mode errors. Third, the laser beam can be delivered by polarization maintaining (PM) fiber to relax the requirement for optical alignment and keep possible heat sources away from the system.
As shown in Fig. 2, in order to eliminate polarization mixing or frequency mixing of a traditional heterodyne laser source, feasibility experiments were run using an optical source with two spatially separated beams driven by two acousto-optic frequency shifters (AOFSs) [13]. Two AOFSs (AOFS1, AOFS2, MT80-B30A1-VIS, A.A.) were employed to generate two acousto-optic frequencies of δf 1 and δf 2 from a stabilized single frequency (f 0 ) source (λ0 632.8 nm, 25-STP-910-230, Melles Griot). The first beam from AOFS1 had frequency f 1 f 0 δf 1 , and the second beam from AOFS2 had frequency f 2 f 0 δf 2 . So beat frequency f s f s f 2 − f 1 f 0 δf 2 − f 0 δf 1 δf 2 − δf 1 was created, and none of the beams had a leakage component to cause periodic nonlinearity. The beams with f 0 and other beams with higher orders (except first order) from AOFS1 and AOFS2 were blocked with pinholes (PHs), which were approximately 200 mm apart from the AOFSs. The two beams from AOFSs were adjusted and aligned before they entered the interferometer. The polarization direction was adjusted by two half-wave plates (HWP1 and HWP2). In an ideal case, the irradiances at PDR and PDM can be expressed as I r ∝ cos2πf 1 t θ1 cos2πf 2 t θ2 ;
(1)
I m ∝ cos2πf 1 t θ1 δ1 θm cos2πf 2 t θ2 δ2 ; (2) where θ1 and θ2 are the phase offsets for AOFS1 and AOFS2, δ1 and δ2 are the phase offsets in the optics paths for measurement and reference beams, and θm is the Doppler phase driven by the measurement mirror. By substituting f s f 2 − f 1 and ignoring the optical frequencies too high to detect, the measured irradiances can be simplified as shown: I r ∝ cos2πf s t θ1 − θ2 ;
(3)
I m ∝ cos2πf s t θ1 –θ2 δ1 –δ2 θm :
(4)
The common-mode errors and thermal errors in the interferometer can be eliminated by designing balanced optics paths for the measurement and reference beams, so that δ1 –δ2 0. The measured
Fig. 2. Optical source with two spatially separated beams driven by two AOFSs at slightly different frequencies from a stabilized source. BS, beam splitter; AOFS1 and AOFS2, acousto-optic frequency shifters; HWP1 and WHP2, half-wave plates; PH, pinhole. 20 August 2014 / Vol. 53, No. 24 / APPLIED OPTICS
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phase difference between I r and I m can be expressed as ϕmr ϕm − ϕr θm . Clearly, the measuring phase after the AOFSs cancels the AOFSs’ differential phase shift. If two spatially separated beams coming out from the optical source are delivered by two PM fibers, the fiber-induced drifts would be included in θ1 and θ2 and canceled in the measuring phase as well. 3. Results and Discussion
As described by Badami and Patterson [14], the actual (nonideal) time-varying component of the PDM signal in a traditional heterodyne interferometer can be expressed as I m ∝ jA1 j2 jA2 j2 jε12 j2 jε21 j2 |{z} (
DCterms
2R A1 A2 ei2πf s tϕm |{z} Base signal
A2 ε21
A1 ε12 ei2πf s t |{z } First harmonic error terms
)
ε12 ε21 ei2πf s t−ϕm |{z} Second harmonic error term
A1 ε21 A2 ε12 e−iϕm ; |{z}
(5)
Quasi-DCterms
where A1 and A2 are the amplitudes of the measurement and reference beams, respectively, and ε12 and ε21 are the deviations from ideal measurement inputs. So the first choice was to directly measure the magnitudes of first and second harmonic errors using the frequency spectrum of output PDM (the so-called frequency domain method) [14,15]. With δf 1 80.00 MHz, δf 2 80.50 MHz, f s 500 kHz, target M2 was moved at a velocity of approximately 11.12 mm/s to yield Doppler shift Δf d 70.30 kHz. The PDM signal was received and then input into a spectrum analyzer (Agilent MSO9254A). As shown in Fig. 3, the original beat frequency is shifted from
500 to 570.30 kHz. There are no recognizable nanoscale periodic errors with noise level ≤ 0.4 nm at first harmonic frequency (500 kHz) or second harmonic frequency (429.70 kHz), which means, compared to several nanometers in a traditional interferometer, the periodic error has been reduced by the proposed interferometer to a subnanometer level. The phase quadrature measurement method was then used to measure the residual periodic errors. As reported by Wu et al. in [11], the relation between periodic phase error (dϕ) and amplitude change (dR∕R) can be expressed as dR : (6) jdϕj R With δf 1 80.00 MHz, δf 2 80.06 MHz, f s 60 kHz, target M2 was moved at 1.4 μm/s to yield Doppler shift Δf d 9 Hz. The PDR signal was received as a reference input, and the PDM signal was received and then demodulated in a digital lock-in amplifier (SR830). The upper limit of the digital lock-in amplifier was 102 kHz (60 kHz < 102 kHz, second harmonic frequency of the detectors 120 kHz > 102 kHz), and so f s 60 kHz was a right choice. The output quadrature components of the digital lock-in amplifier were acquired and offline calculated using MATLAB software. The calculation with MATLAB could be expressed as 8 Rn C2 n S2 n1∕2 ; > > > > ϕn tan−1 Sn∕Cn; > > Rn− ; dϕn dRn > ¯ ¯ R R > > > Filterdϕn; > : λ0 dϕn; dLn 14 2π
and FTdL
X Δn i1
FFTdLi:Δn:N · window
2 ; N (8)
Fig. 3. Periodic errors measured using frequency domain method at first harmonic frequency (500 kHz) and second harmonic frequency (429.70 kHz). 5450
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where n is the index of sample data; N is the number of samples; 4 is number of optical paths to target M2; Cn, Sn are the output quadrature components (cosine, sine) of the digital lock-in amplifier; and dLn is the displacement corresponding to phase dϕn. FFTdLi:Δn:N · window includes the data sampled for the second time at Δn intervals for an appropriate frequency range, the addition of a window function to reduce the fast Fourier transform (FFT) errors, and the calculation of FFT. The purpose of Eq. (8) is to reduce the white noise by doing Δn loops and averaging. FTdL is the result of the Fourier transform of dLn. As shown in Fig. 4, the first harmonic peak amplitude (at 9 Hz) is 36 pm, and the second harmonic peak amplitude (at 18 Hz) is 10 pm. No imperfect optics can cause a periodic error in the proposed interferometer, and the electronic demodulation periodic error of the digital
Fig. 5. Experimental result of thermal error. The error curve (dotted) represents the case with a glass plate 3 mm thick of the same material as PBS1 placed closely between QWP1 and M2; the error curve (solid) represents the other case without the glass plate.
error compared to the unbalanced case. An ideal PBS1 can cancel the slight influence of thermal error while the proposed interferometer is used for displacement measurement. 4. Conclusion
Fig. 4. Periodic errors with 1.4 μm/s from phase quadrature measurement method. (a) Demodulated periodic phase measured with a digital lock-in amplifier; (b) periodic error calculated with 1∕4 λ0 ∕2π dR∕R (4 is number of optical paths to target M2, λ0 632.8 nm is laser wavelength); (c) FFT result of periodic error 4(b). The dominant frequency is approximately 9 Hz, the first harmonic peak amplitude (at 9 Hz) is 36 pm, and the second harmonic peak amplitude (at 18 Hz) is 10 pm.
lock-in amplifier is less than 2.5 pm [11]. The second harmonic frequency of the detectors is out of the range of the digital lock-in amplifier. Shot noise of the detectors and white noise of data acquisition circuit are not periodic and can be eliminated by filtering and averaging. The variation of laser amplitude has nothing to do with the first and second order optical period, and their influence can be minimized by averaging. The periodic stage motion errors and mount vibration errors have nothing to do with the first and second order optical period by choosing the target moving speed. The FFT errors can be reduced by using a window function. Thus, the peak of the periodic error in Fig. 4(c) is attributed to ghost reflection [7]. The immunity of the proposed interferometer from thermal error was investigated with a temperature increment of 1°C per hour from 20°C to 26°C. As shown in Fig. 5, the thermal error achieved with a 3 mm glass plate placed closely between QWP1 and M2 has an approximately linear growth as temperature rises. The thermal error achieved without a 3 mm glass plate goes down to 2 nm, which means the proposed interferometer is immune from thermal
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