- Email: [email protected]
Phase stepping methods based on PTDC for Fiber-Optic Projected-Fringe Digital Interferometry
Optics & Laser Technology 44 (2012) 1089–1094
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Phase stepping methods based on PTDC for Fiber-Optic Projected-Fringe Digital Interferometry Zhang Chao n, Duan Fa-Jie State Key Lab of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
abstract
Article history: Received 8 August 2011 Received in revised form 17 September 2011 Accepted 8 October 2011 Available online 16 November 2011
Active homodyne control can be used to stabilize; p/2-rad phase steps in a Fiber-Optic Projected-Fringe Digital Interferometry. Two beams emitted from a fiber-optic coupler are combined to form an interference fringe pattern on a diffusely reflecting object. Fresnel reflections from the distal fiber ends undergo a double pass in the fibers and interference at the fourth port of the coupler which formed a Michelson interferometer. We suggested a method of PTDC (DC phase tracking) to maintain the interference intensity at quadrature by feedback control. Stepping between quadrature positions force a p/2-rad phase step. A method based on the ratio of harmonic of the interference signal is proposed to estimate phase step accuracy .A root-mean-square phase stability of 2 mrad and phase step accuracy of 13.8 mrad were measured with PTDC for the Fiber-Optic Projected-Fringe Digital Interferometry. It worked well in 2 h without resetting the integrator. & 2011 Elsevier Ltd. All rights reserved.
Keywords: DC phase tracking Active homodyne Phase stepping
1. Introduction Phase-shifting interferometry (PSI) is an important technique in the field of interferometry, providing a tool to measure the optical phase to unprecedented accuracy full-field. Phase-shifting methods may be broadly grouped as temporal and spatial techniques. For temporal algorithms, a discrete or continuous phase shift is introduced. In the phase-stepping technique, the phase is stepped between each intensity measurement, whereas in the continuous phase modulating technique the phase is usually shifted linearly in a saw-tooth like manner or a sinusoidal like manner. Fiber-Optic Projected-Fringe Digital Interferometry has many advantages such as its small size, remote location of the laser source and strong anti-interference ability to electromagnetic radiation and has been used in many areas [1–4]. Automated analysis of interferograms from a Fiber-Optic Projected-Fringe Digital Interferometer by means of phase stepping need the relative phase of the interfering beams to be shifted between the acquisitions of at least three interferograms. In practice the most important type of reasons that may affect the accuracy of the phase measurement technique are errors due to incorrect phase steps between interferograms. Although many algorithms were developed for dealing with arbitrary phase steps [5–7], they need a lot of time and require the interferograms stable. There are algorithms that need the phase steps to be equal
n
Corresponding author. E-mail address: [email protected] (Z. Chao).
0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.10.005
regardless of their correct values; it was proposed by Srinivasan years ago [8]. Most of the algorithms need a precise value of the phase step and consequently the phase modulator must be carefully calibrated to minimize systematic errors. The random phase errors are particularly large in fiber interference systems due to temperature (approximately 106(rad/K)/m). Many stabilized phase-stepping approaches have been developed for this Fiber-Optic Projected-Fringe Digital Interferometry. Quadrant silicon photodiode was used to detect fringe’s phase and tuned the laser wavelength to lock phase by Kudryashov and Seliverstov [9], it reached 0.005 l in rms and 0.0015 l in p–v of reconstruction error. A phase step by sawtooth-current modulation of a laser diode was locked to a phase difference preset by polarization optics through an electrical feedback system; the optical path difference can be precisely measured [10]. Mercer achieved a root-mean-square phase stability of 7.3 mrad in a 40-Hz band-width in a fiber-optic Michelson interferometer by means of PTAC and the phase carrier applied (0.21-rad amplitude) degraded the fringe visibility by approximately 1% [11]. However, phase-stepping error was not measured. Corke and Josten used a feedback system with a piezo regulator which was used for compensation of the phase fluctuations through PTDC [12,13] and kept a given value of the relative phase stable to within 70 mrad [13]. A root-mean-square phase stability of 0.61 mrad in a 50-Hz band-width and phase step accuracy of 1.17 mrad were measured and suggested a mean to estimate phase-stepping error by Moore using PTDC [14] while the system was complicated. In this paper, we used PTDC (DC phase tracking) to compensate the phase fluctuations and made some improvements for
1090
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
electrical feedback system. A method based on the ratio of harmonic of the interference signal was proposed to real-time estimate phase step error.
V (V E1) is the fringe contrast, phase fluctuation for the phase error is fe, the phase-voltage coefficient for PD is kD When the interference intensity is controlled at quadrate
2. Active homodyne phase stabilization with PTDC As Fig. 1 shows, light from a He–Ne laser was coupled into one end of the fiber-optic coupler, polished the output fibers and makes the path difference between the two arms as short as possible. The reflection from the fiber ends constituted a Michelson interferometer. PD detected the interference signal and then interference signal was transmitted to the servo control system. As shown in Fig. 2, servo control system was constructed by pre-chopper-stabilized preamplifier, a low-pass filter with a bandwidth of 500 Hz, comparator (the voltage U0 is described followed), high-pass filter and proportional–integral–derivative (PID) controller. The PID controller’s signal was exported into the high-voltage Piezoelectric Transducer (PZT) driver. As shown in Fig. 3, Photodiode (PD) received signal induced by temperature change is almost 0.1 Hz/K. The cutoff frequency of highpass filter is 0.01 Hz. The high-pass filter introduced here have three advantages: to eliminate the phase fluctuation caused by temperature change, to compensate the change of the DC operating point U0, to overcome the low frequency electronic noise. PD received signal is I ¼ kD ½I0 þ V cosð2ðfo fr þsðtÞÞ þ fe Þ
p
þkpðk ¼ 0, 7 1, 72. . .Þ 2 p ¼ kD ðI0 7 sin fe Þ kD I0 þ kp þ fe I ¼ kD I0 þ cos 2 7 kD fe ðk ¼ 0, 7 1, 72. . .Þ
2ðfd þ sðtÞÞ ¼
ð3Þ
While fo fr ¼ fd, U0 ¼kDI0, the DC operating point is U0. Fig. 4 represents the phase feedback control system model. Fi is the target phase between two output arms, the actual optical
ð1Þ
I0 is the light intensity of background. fo fr is the output fiber phase difference, s(t) is the phase induced by PZT modulated,
ð2Þ
Fig. 3. PD (Photodiode) received signal induced by temperature change.
Fig. 1. Fiber-Optic Projected-Fringe Digital Interferometry with PTDC.
U0
Fig. 2. Servo control system diagram.
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
phase difference is Fo, phase detector output of PD is Fe, PID controller’s transfer function is kc þki/sþkds, voltage-phase coefficient for PZT is kp. System transfer function is HðsÞ ¼
s kd kp s2 þ ð1þ kc kp Þs þki kp
ð4Þ
Dj ¼ j(x,y,t) þ j0, j0 is the phase step for the fringe map, j(x,y,t) represents the phase distribution of the measured object, fx, fy are the space frequency of fringe. While j0 is 0, p/2, p, 3p/2, the fringe map is I2 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ p=2Þ ¼ AB sinðf x u þf y v þ jðx,y,tÞÞ I3 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ pÞ
Error transfer function is EðsÞ ¼
1091
¼ AB cosðf x u þ f y v þ jðx,y,tÞÞ
kd kp s2 þ kc kp s þ ki kp kd kp s2 þ ð1 þkc kp Þs þ ki kp
ð5Þ
I4 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ 3p=2Þ ¼ A þ B sinðf x u þ f y v þ jðx,y,tÞÞ
ð8Þ
Steady-state error is ess ¼ limsEðsÞ ¼ 0
ð6Þ
s-0
As shown in Fig. 5, PD received signal when PZT is modulated by triangular wave, channel 2 showing is modulating signal and the signal PD received is the channel 1. As the modulating voltage changes from 1.5 V to 1.5 V (the high-voltage PZT driver’s amplification coefficient is 24), the phase change is 3p (for the Michelson interferometer) and the voltage-phase coefficient for PZT kp is 0.131 rad/V. We also can see kD (the phase-voltage coefficient for PD) is 0.5 V/rad and the DC operating point U0. After closed loop PID controlling, adjusted parameters, the system reached steady state. Fig. 6 is the phase controlling signal before and after the phase stabling. It can be seen the PD received signal’s fluctuation is less than 2 mV after phase control, according to kD is 0.5 V/rad; the phase fluctuation are less than 4 mrad for the Michelson interferometer and 2 mrad for the fringe map.
3. Phase stepping and estimating phase-stepping error 3.1. Phase stepping Stepping between quadrature positions (p phase step) in Michelson interferometer force a p/2-rad phase step for the fringe map. The fringe map is displayed as Iðx,y,tÞ ¼ A þB cosðf x x þ f y y þ DjÞ
Φi
f x u þ f y u þ jðx,y,tÞ ¼ arctan
I4 I2 I1 I3
ð9Þ
From the reference map, we can get f x u þ f y v ¼ arctan
I40 I20 I10 I30
ð10Þ
We get the phase distribution of the measured object and the phase distribution of the reference object after phase unwrapping. Phase-stepping method: 1. Modulate PZT with triangular wave and get the value of kp and the DC operating point U0 as described in Fig. 5. 2. With the help of kp and U0, get the phase step reference voltage U1, U2, and U3. Stepping between quadrature positions (p phase step) in Michelson interferometer force a p/2-rad phase step for the fringe map U1 ¼ U0 þ
p kp
ð11Þ
U2 ¼ U0 þ
2p kp
ð12Þ
U3 ¼ U0 þ
3p kp
ð13Þ
3. Make U0, U1, U2, U3 as the phase step reference voltage, close the closed-loop feedback system and change the polarity of the feedback loop, let the phase stable.
ð7Þ
Φo
Φe
k
Fig. 4. Phase feedback control system model.
Fig. 7 is the phase stepping process with PTDC. The phase steps are: p, 2p, p for the Michelson interferometer and p/2, p, p/2 for the fringe map. The PD received signal’s fluctuation is less than 2 mV within the phase stepping process. 3.2. Estimating phase-stepping error After closed the closed-loop feedback system, when PZT1 is in small amplitude sinusoidal modulating, the interference signal
Fig. 5. The signal PD received while PZT is modulated with triangular wave.
1092
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
Fig. 6. The signal of the PD before and after phase controlling. (a) The signal before phase controlling. (b) The signal after phase controlling. (c) Before and after phase controlling. (d) Fluctuation signal when the phase stabilized.
the phase error is fe, z is the amplitude of the phase modulation signal, o is the frequency of the phase modulation signal and y is the initial phase of the phase modulation signal. Jn(z) is the n-order Bessel coefficient. When the interference intensity is controlled at quadrate for PTDC: a ¼(p/2)þkp(k¼0, 71, y): The first harmonic is Sh1 ðtÞ ¼ 2B sinða þ fe ÞJ 1 ðzÞcosðot þ yÞ
ð15Þ
The second harmonic is Sh2 ðtÞ ¼ 2B cosða þ fe ÞJ2 ðzÞcosð2ot þ 2yÞ
ð16Þ
The coefficients of the first harmonic and the second harmonic are
Fig. 7. Phase stepping process with PTDC.
K P1 ¼ 2B sinða þ fe ÞJ 1 ðzÞ
ð17Þ
K P2 ¼ 2B cosða þ fe ÞJ 2 ðzÞ
ð18Þ
For the k-order Bessel coefficient Jk(z) before PD is described as Jk ðzÞ ¼
SðtÞ ¼ A þ B cos½z cosðot þ yÞ þ a þ fe
1 X
ð1Þm
m¼0
¼ A þ B cosða þ fe Þ½J0 ðzÞ2J2 ðzÞcosð2ot þ2yÞ þ . . .
z k þ 2m 1 m!ðk þmÞ! 2
ð19Þ
While PZT1 is in small amplitude sinusoidal phase modulating
B sinða þ fe Þ½2J 1 ðzÞcosðot þ yÞ2J3 ðzÞcosð3ot þ 3yÞ. . . ð14Þ A is the light intensity of background. a is the desired output fiber phase difference, B(BE1) is the fringe contrast, phase fluctuations for
J1 ðzÞ ¼
1 X m¼0
ð1Þm
z 1 þ 2m 1 z 1 z 3 1 z 5 z ¼ þ ... m!ð1þ mÞ! 2 2 1!2! 2 2!3! 2 2
ð20Þ
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
1093
Fig. 8. Phase-stepping errors for PTDC. (a) Sinusoidal phase modulating of the amplitude 2 V. (b) PD received signal after phase control. (c) The frequency spectrum of (b). (d) The frequency spectrum of the signal of (b) passed a high-pass filter of the cutoff frequency 60 Hz.
J 2 ðzÞ ¼
z 2 þ 2m 1 z 2 1 1 z 4 ¼ m!ð2 þmÞ! 2 2 2 1!3! 2 m¼0 1 z 6 1 z 2 þ ... 2!4! 2 2 2 1 X
ð1Þm
ð21Þ
We can get K P1 ¼ 2B sinða þ fe Þ
z ¼ B sinða þ fe Þz 2
K P2 ¼ 2B cosða þ fe ÞJ 2 ðzÞ ¼ 2B cosða þ fe Þ ¼
K PTDC ¼
ð22Þ
Optic Projected-Fringe Digital Interferometry. Stepping between quadrature positions (p phase step) for the optical fiber Michelson interferometer force a p/2-rad phase step. A method based on the ratio of harmonic of the interference signal was proposed to estimate phase step accuracy. The phase fluctuation were less than 4 mrad for the Michelson interferometer and 2 mrad for the fringe map, the phase step error were less than 27.5 mrad for the Michelson interferometer and 13.8 mrad for the fringe map were measured. It worked well in 2 h without resetting the integrator.
1 z 2 2 2
B cosða þ fe Þz2 4
ð23Þ
Acknowledgments
K P2 z ¼ ctgða þ fe Þ 4 K P1
ð24Þ
This paper is supported by the State Key Lab Explore Fund and The National Natural Science Foundation of China (NSFC): 50375110.
ð25Þ
References
p 4K PTDC þ kpafe ¼ tanðfe Þ fe ¼ ctgða þ fe Þ ¼ tan 2 z
Fig. 8 shows the phase-stepping error for PTDC. Sinusoidal phase modulating of the amplitude for PZT1 is 2 V and the voltage-phase coefficient for PZT1 kp is 0.131 rad/V (in Fig. 5).We can get z¼0.262 rad.The value of KPTDC is superior to 55 dB (in Fig. 7(d)). From formula (23), we get the phase step error are less than 27.5 mrad for the Michelson interferometer and 13.8 mrad for the fringe map.
4. Conclusion We used a method of PTDC (DC phase tracking) to maintain the interference intensity at quadrature by feedback control for Fiber-
[1] Spagnolo Giuseppe Schirripa. Vibtation monitoring by fiber optic fringe projection and Fourier transform analysis. Optics Communications 1997;139: 17–23. [2] Ton Timothy LPenning, Xiao Hai, May Russell, Wang Anbo. Miniaturized 3-D surface profilometry using fiber optic coupler. Optics & Laser Technology 2001;33:313–20. [3] Xiao Hai, Zhang Yibing. Anbo Wang. Multi-spectral three-dimensional digital infrared thermal imaging. Optical Engineering 2003;42(4):906–11. [4] Wu Fang, Zhang Hong, Labor Michael J, Burton David R. A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry. Optics Communications 2001;187:347–57. [5] Chen Xin, Gramaglia Maureen. Phase-shifting interferometry with an calibrated phase shifts. Applied Optics 2002;39:585–91. [6] Wang Zhaoyang, Han Bonita. Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms. Optics Letters 2004;29: 1671–3.
1094
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
[7] Xu J, Xu Q, Chai L. Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts. Applied Optics 2008;47:480–502. [8] Srinivasan V. Automated phase-measuring profilometry of 3-D diffuse objects. Applied Optics 1987;23(18):3105–8. [9] Kudryashov Alexis V, Seliverstov Aleksei V. Adaptive stabilized interferometer with laser diode. Optics Communications 1995;120:239–44. [10] Onodera Ribun. Phase-shift-locked interferometer with a wavelength-modulated laser diode. Applied Optics 2003;42(1):91–6.
[11] Mercer Carolyn R, Beheim Glenn. Fiber optic phase stepping system for interferometry. Applied Optics 1991;30(7):729–34. [12] Corke M, Jones JDC. All single-mode fibre optic holographic system with active fringe stabilization. Journal of Physics E: Scientific Instruments 1985;18:185–6. [13] Josten G. Active phase stabilization in a two-fiber interferometer. Applied Physics B 1990;51:418–20. [14] Moore Andrew J, McBride Roy. Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement. Applied Optics 2002;41: 3348–54.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Phase stepping methods based on PTDC for Fiber-Optic Projected-Fringe Digital Interferometry Zhang Chao n, Duan Fa-Jie State Key Lab of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China
a r t i c l e i n f o
abstract
Article history: Received 8 August 2011 Received in revised form 17 September 2011 Accepted 8 October 2011 Available online 16 November 2011
Active homodyne control can be used to stabilize; p/2-rad phase steps in a Fiber-Optic Projected-Fringe Digital Interferometry. Two beams emitted from a fiber-optic coupler are combined to form an interference fringe pattern on a diffusely reflecting object. Fresnel reflections from the distal fiber ends undergo a double pass in the fibers and interference at the fourth port of the coupler which formed a Michelson interferometer. We suggested a method of PTDC (DC phase tracking) to maintain the interference intensity at quadrature by feedback control. Stepping between quadrature positions force a p/2-rad phase step. A method based on the ratio of harmonic of the interference signal is proposed to estimate phase step accuracy .A root-mean-square phase stability of 2 mrad and phase step accuracy of 13.8 mrad were measured with PTDC for the Fiber-Optic Projected-Fringe Digital Interferometry. It worked well in 2 h without resetting the integrator. & 2011 Elsevier Ltd. All rights reserved.
Keywords: DC phase tracking Active homodyne Phase stepping
1. Introduction Phase-shifting interferometry (PSI) is an important technique in the field of interferometry, providing a tool to measure the optical phase to unprecedented accuracy full-field. Phase-shifting methods may be broadly grouped as temporal and spatial techniques. For temporal algorithms, a discrete or continuous phase shift is introduced. In the phase-stepping technique, the phase is stepped between each intensity measurement, whereas in the continuous phase modulating technique the phase is usually shifted linearly in a saw-tooth like manner or a sinusoidal like manner. Fiber-Optic Projected-Fringe Digital Interferometry has many advantages such as its small size, remote location of the laser source and strong anti-interference ability to electromagnetic radiation and has been used in many areas [1–4]. Automated analysis of interferograms from a Fiber-Optic Projected-Fringe Digital Interferometer by means of phase stepping need the relative phase of the interfering beams to be shifted between the acquisitions of at least three interferograms. In practice the most important type of reasons that may affect the accuracy of the phase measurement technique are errors due to incorrect phase steps between interferograms. Although many algorithms were developed for dealing with arbitrary phase steps [5–7], they need a lot of time and require the interferograms stable. There are algorithms that need the phase steps to be equal
n
Corresponding author. E-mail address: [email protected] (Z. Chao).
0030-3992/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2011.10.005
regardless of their correct values; it was proposed by Srinivasan years ago [8]. Most of the algorithms need a precise value of the phase step and consequently the phase modulator must be carefully calibrated to minimize systematic errors. The random phase errors are particularly large in fiber interference systems due to temperature (approximately 106(rad/K)/m). Many stabilized phase-stepping approaches have been developed for this Fiber-Optic Projected-Fringe Digital Interferometry. Quadrant silicon photodiode was used to detect fringe’s phase and tuned the laser wavelength to lock phase by Kudryashov and Seliverstov [9], it reached 0.005 l in rms and 0.0015 l in p–v of reconstruction error. A phase step by sawtooth-current modulation of a laser diode was locked to a phase difference preset by polarization optics through an electrical feedback system; the optical path difference can be precisely measured [10]. Mercer achieved a root-mean-square phase stability of 7.3 mrad in a 40-Hz band-width in a fiber-optic Michelson interferometer by means of PTAC and the phase carrier applied (0.21-rad amplitude) degraded the fringe visibility by approximately 1% [11]. However, phase-stepping error was not measured. Corke and Josten used a feedback system with a piezo regulator which was used for compensation of the phase fluctuations through PTDC [12,13] and kept a given value of the relative phase stable to within 70 mrad [13]. A root-mean-square phase stability of 0.61 mrad in a 50-Hz band-width and phase step accuracy of 1.17 mrad were measured and suggested a mean to estimate phase-stepping error by Moore using PTDC [14] while the system was complicated. In this paper, we used PTDC (DC phase tracking) to compensate the phase fluctuations and made some improvements for
1090
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
electrical feedback system. A method based on the ratio of harmonic of the interference signal was proposed to real-time estimate phase step error.
V (V E1) is the fringe contrast, phase fluctuation for the phase error is fe, the phase-voltage coefficient for PD is kD When the interference intensity is controlled at quadrate
2. Active homodyne phase stabilization with PTDC As Fig. 1 shows, light from a He–Ne laser was coupled into one end of the fiber-optic coupler, polished the output fibers and makes the path difference between the two arms as short as possible. The reflection from the fiber ends constituted a Michelson interferometer. PD detected the interference signal and then interference signal was transmitted to the servo control system. As shown in Fig. 2, servo control system was constructed by pre-chopper-stabilized preamplifier, a low-pass filter with a bandwidth of 500 Hz, comparator (the voltage U0 is described followed), high-pass filter and proportional–integral–derivative (PID) controller. The PID controller’s signal was exported into the high-voltage Piezoelectric Transducer (PZT) driver. As shown in Fig. 3, Photodiode (PD) received signal induced by temperature change is almost 0.1 Hz/K. The cutoff frequency of highpass filter is 0.01 Hz. The high-pass filter introduced here have three advantages: to eliminate the phase fluctuation caused by temperature change, to compensate the change of the DC operating point U0, to overcome the low frequency electronic noise. PD received signal is I ¼ kD ½I0 þ V cosð2ðfo fr þsðtÞÞ þ fe Þ
p
þkpðk ¼ 0, 7 1, 72. . .Þ 2 p ¼ kD ðI0 7 sin fe Þ kD I0 þ kp þ fe I ¼ kD I0 þ cos 2 7 kD fe ðk ¼ 0, 7 1, 72. . .Þ
2ðfd þ sðtÞÞ ¼
ð3Þ
While fo fr ¼ fd, U0 ¼kDI0, the DC operating point is U0. Fig. 4 represents the phase feedback control system model. Fi is the target phase between two output arms, the actual optical
ð1Þ
I0 is the light intensity of background. fo fr is the output fiber phase difference, s(t) is the phase induced by PZT modulated,
ð2Þ
Fig. 3. PD (Photodiode) received signal induced by temperature change.
Fig. 1. Fiber-Optic Projected-Fringe Digital Interferometry with PTDC.
U0
Fig. 2. Servo control system diagram.
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
phase difference is Fo, phase detector output of PD is Fe, PID controller’s transfer function is kc þki/sþkds, voltage-phase coefficient for PZT is kp. System transfer function is HðsÞ ¼
s kd kp s2 þ ð1þ kc kp Þs þki kp
ð4Þ
Dj ¼ j(x,y,t) þ j0, j0 is the phase step for the fringe map, j(x,y,t) represents the phase distribution of the measured object, fx, fy are the space frequency of fringe. While j0 is 0, p/2, p, 3p/2, the fringe map is I2 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ p=2Þ ¼ AB sinðf x u þf y v þ jðx,y,tÞÞ I3 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ pÞ
Error transfer function is EðsÞ ¼
1091
¼ AB cosðf x u þ f y v þ jðx,y,tÞÞ
kd kp s2 þ kc kp s þ ki kp kd kp s2 þ ð1 þkc kp Þs þ ki kp
ð5Þ
I4 ðx,y,tÞ ¼ A þ B cosðf x u þ f y v þ jðx,y,tÞ þ 3p=2Þ ¼ A þ B sinðf x u þ f y v þ jðx,y,tÞÞ
ð8Þ
Steady-state error is ess ¼ limsEðsÞ ¼ 0
ð6Þ
s-0
As shown in Fig. 5, PD received signal when PZT is modulated by triangular wave, channel 2 showing is modulating signal and the signal PD received is the channel 1. As the modulating voltage changes from 1.5 V to 1.5 V (the high-voltage PZT driver’s amplification coefficient is 24), the phase change is 3p (for the Michelson interferometer) and the voltage-phase coefficient for PZT kp is 0.131 rad/V. We also can see kD (the phase-voltage coefficient for PD) is 0.5 V/rad and the DC operating point U0. After closed loop PID controlling, adjusted parameters, the system reached steady state. Fig. 6 is the phase controlling signal before and after the phase stabling. It can be seen the PD received signal’s fluctuation is less than 2 mV after phase control, according to kD is 0.5 V/rad; the phase fluctuation are less than 4 mrad for the Michelson interferometer and 2 mrad for the fringe map.
3. Phase stepping and estimating phase-stepping error 3.1. Phase stepping Stepping between quadrature positions (p phase step) in Michelson interferometer force a p/2-rad phase step for the fringe map. The fringe map is displayed as Iðx,y,tÞ ¼ A þB cosðf x x þ f y y þ DjÞ
Φi
f x u þ f y u þ jðx,y,tÞ ¼ arctan
I4 I2 I1 I3
ð9Þ
From the reference map, we can get f x u þ f y v ¼ arctan
I40 I20 I10 I30
ð10Þ
We get the phase distribution of the measured object and the phase distribution of the reference object after phase unwrapping. Phase-stepping method: 1. Modulate PZT with triangular wave and get the value of kp and the DC operating point U0 as described in Fig. 5. 2. With the help of kp and U0, get the phase step reference voltage U1, U2, and U3. Stepping between quadrature positions (p phase step) in Michelson interferometer force a p/2-rad phase step for the fringe map U1 ¼ U0 þ
p kp
ð11Þ
U2 ¼ U0 þ
2p kp
ð12Þ
U3 ¼ U0 þ
3p kp
ð13Þ
3. Make U0, U1, U2, U3 as the phase step reference voltage, close the closed-loop feedback system and change the polarity of the feedback loop, let the phase stable.
ð7Þ
Φo
Φe
k
Fig. 4. Phase feedback control system model.
Fig. 7 is the phase stepping process with PTDC. The phase steps are: p, 2p, p for the Michelson interferometer and p/2, p, p/2 for the fringe map. The PD received signal’s fluctuation is less than 2 mV within the phase stepping process. 3.2. Estimating phase-stepping error After closed the closed-loop feedback system, when PZT1 is in small amplitude sinusoidal modulating, the interference signal
Fig. 5. The signal PD received while PZT is modulated with triangular wave.
1092
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
Fig. 6. The signal of the PD before and after phase controlling. (a) The signal before phase controlling. (b) The signal after phase controlling. (c) Before and after phase controlling. (d) Fluctuation signal when the phase stabilized.
the phase error is fe, z is the amplitude of the phase modulation signal, o is the frequency of the phase modulation signal and y is the initial phase of the phase modulation signal. Jn(z) is the n-order Bessel coefficient. When the interference intensity is controlled at quadrate for PTDC: a ¼(p/2)þkp(k¼0, 71, y): The first harmonic is Sh1 ðtÞ ¼ 2B sinða þ fe ÞJ 1 ðzÞcosðot þ yÞ
ð15Þ
The second harmonic is Sh2 ðtÞ ¼ 2B cosða þ fe ÞJ2 ðzÞcosð2ot þ 2yÞ
ð16Þ
The coefficients of the first harmonic and the second harmonic are
Fig. 7. Phase stepping process with PTDC.
K P1 ¼ 2B sinða þ fe ÞJ 1 ðzÞ
ð17Þ
K P2 ¼ 2B cosða þ fe ÞJ 2 ðzÞ
ð18Þ
For the k-order Bessel coefficient Jk(z) before PD is described as Jk ðzÞ ¼
SðtÞ ¼ A þ B cos½z cosðot þ yÞ þ a þ fe
1 X
ð1Þm
m¼0
¼ A þ B cosða þ fe Þ½J0 ðzÞ2J2 ðzÞcosð2ot þ2yÞ þ . . .
z k þ 2m 1 m!ðk þmÞ! 2
ð19Þ
While PZT1 is in small amplitude sinusoidal phase modulating
B sinða þ fe Þ½2J 1 ðzÞcosðot þ yÞ2J3 ðzÞcosð3ot þ 3yÞ. . . ð14Þ A is the light intensity of background. a is the desired output fiber phase difference, B(BE1) is the fringe contrast, phase fluctuations for
J1 ðzÞ ¼
1 X m¼0
ð1Þm
z 1 þ 2m 1 z 1 z 3 1 z 5 z ¼ þ ... m!ð1þ mÞ! 2 2 1!2! 2 2!3! 2 2
ð20Þ
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
1093
Fig. 8. Phase-stepping errors for PTDC. (a) Sinusoidal phase modulating of the amplitude 2 V. (b) PD received signal after phase control. (c) The frequency spectrum of (b). (d) The frequency spectrum of the signal of (b) passed a high-pass filter of the cutoff frequency 60 Hz.
J 2 ðzÞ ¼
z 2 þ 2m 1 z 2 1 1 z 4 ¼ m!ð2 þmÞ! 2 2 2 1!3! 2 m¼0 1 z 6 1 z 2 þ ... 2!4! 2 2 2 1 X
ð1Þm
ð21Þ
We can get K P1 ¼ 2B sinða þ fe Þ
z ¼ B sinða þ fe Þz 2
K P2 ¼ 2B cosða þ fe ÞJ 2 ðzÞ ¼ 2B cosða þ fe Þ ¼
K PTDC ¼
ð22Þ
Optic Projected-Fringe Digital Interferometry. Stepping between quadrature positions (p phase step) for the optical fiber Michelson interferometer force a p/2-rad phase step. A method based on the ratio of harmonic of the interference signal was proposed to estimate phase step accuracy. The phase fluctuation were less than 4 mrad for the Michelson interferometer and 2 mrad for the fringe map, the phase step error were less than 27.5 mrad for the Michelson interferometer and 13.8 mrad for the fringe map were measured. It worked well in 2 h without resetting the integrator.
1 z 2 2 2
B cosða þ fe Þz2 4
ð23Þ
Acknowledgments
K P2 z ¼ ctgða þ fe Þ 4 K P1
ð24Þ
This paper is supported by the State Key Lab Explore Fund and The National Natural Science Foundation of China (NSFC): 50375110.
ð25Þ
References
p 4K PTDC þ kpafe ¼ tanðfe Þ fe ¼ ctgða þ fe Þ ¼ tan 2 z
Fig. 8 shows the phase-stepping error for PTDC. Sinusoidal phase modulating of the amplitude for PZT1 is 2 V and the voltage-phase coefficient for PZT1 kp is 0.131 rad/V (in Fig. 5).We can get z¼0.262 rad.The value of KPTDC is superior to 55 dB (in Fig. 7(d)). From formula (23), we get the phase step error are less than 27.5 mrad for the Michelson interferometer and 13.8 mrad for the fringe map.
4. Conclusion We used a method of PTDC (DC phase tracking) to maintain the interference intensity at quadrature by feedback control for Fiber-
[1] Spagnolo Giuseppe Schirripa. Vibtation monitoring by fiber optic fringe projection and Fourier transform analysis. Optics Communications 1997;139: 17–23. [2] Ton Timothy LPenning, Xiao Hai, May Russell, Wang Anbo. Miniaturized 3-D surface profilometry using fiber optic coupler. Optics & Laser Technology 2001;33:313–20. [3] Xiao Hai, Zhang Yibing. Anbo Wang. Multi-spectral three-dimensional digital infrared thermal imaging. Optical Engineering 2003;42(4):906–11. [4] Wu Fang, Zhang Hong, Labor Michael J, Burton David R. A novel design for fiber optic interferometric fringe projection phase-shifting 3-D profilometry. Optics Communications 2001;187:347–57. [5] Chen Xin, Gramaglia Maureen. Phase-shifting interferometry with an calibrated phase shifts. Applied Optics 2002;39:585–91. [6] Wang Zhaoyang, Han Bonita. Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms. Optics Letters 2004;29: 1671–3.
1094
Z. Chao, D. Fa-Jie / Optics & Laser Technology 44 (2012) 1089–1094
[7] Xu J, Xu Q, Chai L. Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts. Applied Optics 2008;47:480–502. [8] Srinivasan V. Automated phase-measuring profilometry of 3-D diffuse objects. Applied Optics 1987;23(18):3105–8. [9] Kudryashov Alexis V, Seliverstov Aleksei V. Adaptive stabilized interferometer with laser diode. Optics Communications 1995;120:239–44. [10] Onodera Ribun. Phase-shift-locked interferometer with a wavelength-modulated laser diode. Applied Optics 2003;42(1):91–6.
[11] Mercer Carolyn R, Beheim Glenn. Fiber optic phase stepping system for interferometry. Applied Optics 1991;30(7):729–34. [12] Corke M, Jones JDC. All single-mode fibre optic holographic system with active fringe stabilization. Journal of Physics E: Scientific Instruments 1985;18:185–6. [13] Josten G. Active phase stabilization in a two-fiber interferometer. Applied Physics B 1990;51:418–20. [14] Moore Andrew J, McBride Roy. Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement. Applied Optics 2002;41: 3348–54.