Sinusoidal phase modulating interferometry system for 3D profile measurement

Optics & Laser Technology 59 (2014) 137–142

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Sinusoidal phase modulating interferometry system for 3D profile measurement Bo En, Duan Fa-jie n, Lv Chang-rong, Zhang Fu-kai, Feng Fan State Key Lab of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China

art ic l e i nf o

a b s t r a c t

Article history: Received 10 October 2013 Received in revised form 28 November 2013 Accepted 29 November 2013 Available online 15 January 2014

We describe a fiber-optic sinusoidal phase modulating (SPM) interferometer for three-dimensional (3D) profilometry, which is insensitive to external disturbances such as mechanical vibration and temperature fluctuation. Sinusoidal phase modulation is created by modulating the drive voltage of the piezoelectric transducer (PZT) with a sinusoidal wave. The external disturbances that cause phase drift in the interference signal and decrease measuring accuracy are effectively eliminated by building a closed-loop feedback system. The phase stability can be measured with a precision of 2.75 mrad, and the external disturbances can be reduced to 53.43 mrad for the phase of fringe patterns. By measuring the dynamic deformation of the rubber membrane, the RMSE is about 0.018 mm, and a single measurement takes less than 250 ms. The feasibility for real-time application has been verified. & 2013 Elsevier Ltd. All rights reserved.

Keywords: 3D profilometry Fiber-optic interference SPM

1. Introduction With the excellence in high speed and high accuracy, phase profilometry has been exhaustively studied and widely applied for 3D sensing, machine vision, robot simulation, industrial monitoring, dressmaking, biomedicine, and other fields [1]. Several typical 3D profilometry are based on phase techniques, including Fourier transformation profilometry (FTP) [2,3], phase measuring profilometry (PMP) [4,5], moiré technique (MT) [6,7], modulation measurement profilometry (MMP) [8], and so on. Among the phase techniques, the SPM interferometry has a good performance in eliminating the external disturbances, and can easily be performed and operated. The simple configuration of the sinusoidal phase modulation enables us to use an optical fiber effectively as a modulating and transmitting device. Some researchers change the wavelength of the laser diode (LD) by varying the injection current (IC) to realize the SPM [9,10]. However, the light intensity of the LD is also modulated by the variation of the IC, which results in measurement errors and decreases the signal-to-noise ratio of the interference signal. In this paper, a novel all-fiber SPM interferometer with a good suppression property for the external disturbances is proposed. This system is based on SPM technique and makes use of the Mach– Zehnder interferometer structure and Young's double pinhole interference principle to achieve the high-density interference fringe projection. Instead of varying the IC, some optical element such as PZT has been used for SPM, and measurement accuracy is

n

Corresponding author. E-mail address: [email protected] (D. Fa-jie).

0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.11.024

not affected by an intensity modulation that usually appears in the current modulated LD. Researchers extract the phase by sampling the interference fringe pattern on certain condition with a CCD image sensor, and optimize the SPM depth to minimize the effect of the additive noise, as reported in Ref. [9]. Moreover, the initial phase of the interference signal, which is caused by the initial optical path difference between interference arms, can be demodulated using phase generated carrier (PGC) [11] method and counted out using coordinated rotation digital computer (CORDIC) [12,13]. Then a compensation voltage is generated for the PZT driver and a closed-loop compensation system is built. In Section 2, the optical setup of fringe projection is presented, and geometry model is built. A simple resume of the integratingbucket method has been presented. In Section 3, the initial phase of Michelson interference signal is demodulated by PGC method, and calculated by CORDIC. The principle of CORDIC has been presented briefly, and the quantization error of CORDIC is discussed. In Section 4, we verify the performance of phase stabilization system, and the feasibility for profile reconstruction has been demonstrated by measuring the dynamic deformation of rubber membrane.

2. Composition and measuring principle 2.1. Optical setup The optical setup is shown in Fig. 1. A He–Ne laser with a good coherence is chosen to be the light source. The output beam is projected into the first port of a 2
2 optical coupler, and split into reference beam in arm (2) and signal beam in arm (3). The optical

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coupler and optical fiber arm (2) and (3) compose a Mach– Zehnder interferometer, and satisfy Young's double pinhole interference condition. Its fiber end faces are clamped together and its core distance is l. The high-density interference fringe projection of cosine distribution can be obtained when satisfying far-field and paraxial condition. Due to the natural property that the fiber is sensitive to temperature and vibration, the phase of interference fringe will drift. Two Fresnel reflection signals on the two fiber end faces of arm (2) and arm (3) undergo a double pass in the fiber, and form the Michelson interference at the fourth port of the optical coupler. The light signal transmitted out of arm (4) is projected onto PD, and utilized to count out the initial phase in PSS. A compensation voltage is generated for PZT driver. By changing the length of optical fiber arm (3), the external disturbances can be compensated and a stabilized interference fringe will be achieved. The CCD captures the deformed fringe pattern modulated by the object, and finishes the profile reconstruction with some algorithm.

2.2. Measuring principle The geometry model is shown in Fig. 2. The optical center of the camera's lens is selected as the geometry model's origin. The xaxis is paralleled to the camera's horizontal direction, the z-axis is paralleled to the camera's vertical direction, and the y-axis is along the camera's optical axis, respectively. The fiber-optic projector is located at O′(L,0,0), away from the origin O with a distance L. The projected fringes are paralleled to the z-axis. Assuming S(x,y,z) is one point on the screen with the projection angle β, and the

geometry equation of the point is given by x sin β þ z cos β ¼ L sin β

ð1Þ

Si(m,n) is the mapped point on the camera's image plane, where m is the horizontal coordinate and n is the vertical coordinate, respectively. The geometry equation between S and Si can be denoted as x y z ¼ ¼ m d n

ð2Þ

where d is the distance between lens center and the image plane. Solving Eqs. (1) and (2) for (x,y,z), we obtain 8 > x ¼ m  mL > d cot β > < y ¼ m  dnLcot β ð3Þ > > dL > :z ¼  m  d cot β

The distribution of the phase difference between reference arm (2) and signal arm (3) is denoted by φ(x,y), and its relationship with projection angle β can be represented by the following equation: φðx; yÞ ¼

2π l tan ðβ  β0 Þ þφ0 ðx; yÞ λ

ð4Þ

where λ is the wavelength of laser light, l is the core distance between two fiber end faces, β0 is the projection angle between 0-th order fringe and x-axis, and φ0 is the initial phase difference between two interference optical fiber arms. Satisfying the far-field and paraxial conditions, the projected fringe is paralleled to z-axis and Eq. (4) can be simplified as φðx; yÞ ¼

2π lðβ β0 Þ þ φ0 λ

ð5Þ

If we get the phase φ(x,y), then the projected angle β can be solved. Put it in Eq. (3), the 3D profile information (x,y,z) can be achieved. In next section, the sinusoidal phase integrating bucket modulation is shown to extract the phase φ(x,y).

Fig. 1. Optical setup of measuring system. CL, coupled lens; OC, optical coupler; PSS, phase stabilization system. Fig. 3. Schematic of phase stabilization with PGC method.

Fig. 2. Geometry model of measuring system.

Fig. 4. Phase error line of CORDIC.

B. En et al. / Optics & Laser Technology 59 (2014) 137–142

2.3. Sinusoidal phase integrating bucket modulation

represented by

In Fig. 1, the optical fiber signal arm (3) is phase modulated with a sinusoidally vibrating PZT, and its driving signal can be

MðtÞ ¼ a cos ðωt þ θÞ

139

ð6Þ

where a is the amplitude of modulated signal and ω is the modulation frequency, respectively.θ is the initial phase of modulated signal. The time-varying intensity on the CCD image sensor, which is produced by Young's two pinhole, is given by sðx; y; tÞ ¼ A þ B cos ½z cos ðωt þ θÞ þ φðx; yÞ

ð7Þ

where A is the background intensity and B is the contrast between light and dark fringes, respectively. z is phase modulation depth. The time-varying signal s(x,y,t) is integrated with a CCD image sensor in parallel over the four quarters of the modulation period T( ¼2π/ω). These four separate images are given by Ep ¼

4 T

Z

ðpT=4Þ

sðx; y; tÞdt ððp  1ÞT=4Þ

ðp ¼ 1  4Þ

ð8Þ

Considering the influence of the additive noise in the system, the phase φ(x,y) is obtained with the simplified formula under the optimum SPM condition [9] that z¼ 2.45 and θ¼0.96 rad: Fig. 5. The voltage-phase coefficient calibration. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

 φðx; yÞ ¼ arctan

E 1  E2  E 3 þ E 4 E 1  E2 þ E 3  E 4

 ð9Þ

Fig. 6. (a) S(t) without phase compensation, (b) S(t) with phase compensation, (c) demodulated phase α(t), (d) frequency spectrum of S(t). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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second harmonic components F1(t) and F2(t), respectively: ( F 1 ðtÞ ¼ S1 ðtÞnhBPF1 ðtÞ ¼  2D sin ½αðtÞJ 1 ðzÞ cos ðωt þ θÞ F 2 ðtÞ ¼ S1 ðtÞnhBPF2 ðtÞ ¼  2D cos ½αðtÞJ 2 ðzÞ cos ð2ωt þ 2θÞ

3. Stabilization of interference fringe 3.1. PGC phase demodulation The Michelson interference signal detected by PD is given by SðtÞ ¼ C þ D cos ½αðtÞJ 0 ðzÞ p

þ 2D cos ½αðtÞÞ ∑ ð  1Þn J 2n ðzÞ cos ½2nðωt þ θÞ n¼1 p

 2D sin ½αðtÞ ∑ ð 1Þn J 2n þ 1 ðzÞ cos ½ð2n þ 1Þðωt þ θÞ n¼0

ð10Þ

where C represents the background intensity and D represents the modulated intensity of the Michelson interferometer which derives from the Fresnel reflection. α(t) is a low-frequency signal, which is mostly affected by temperature fluctuation, and α0 ¼2φ0. Jn is the first kind of Bessel function. PGC phase demodulation method is mainly composed of three steps: phase generated carrier, detection and phase demodulation. In the first step, the fundamental carrier with frequency ω and the second-harmonic carrier with frequency 2ω are generated and given by X1(t) and X2(t): ( X 1 ðtÞ ¼ G1 cos ðωt þ θÞ ð11Þ X 2 ðtÞ ¼ G2 cos ½2ðωt þ θÞ where G1 and G2 are constants. In the second step, S(t) gets through a band-pass filter BPF1 with ω as the center frequency, and BPF2 with 2ω as the center frequency. We obtain the first and

ð12Þ

where * stands for the convolution operation, hBPF1 ðtÞ and hBPF2 ðtÞ are response functions of BPF1 and BPF2. From Eq. (16), the ratio of first and second harmonic amplitude is given by Fðα; zÞ ¼ tan ðαÞ U J 1 ðzÞ=J 2 ðzÞ

ð13Þ

The harmonic components F1(t) and F2(t) are multiplied by the carriers X1(t) and X2(t), respectively. Let the signals pass a low-pass filter to obtain the V1(t) and V1(t), which contain α(t): ( V 1 ðtÞ ¼ ½F 1 ðtÞ UX 1 ðtÞnhLPF1 ðtÞ ¼  DG1 sin ½αðtÞJ 1 ðzÞ ð14Þ V 2 ðtÞ ¼ ½F 2 ðtÞ UX 2 ðtÞnhLPF2 ðtÞ ¼  DG2 cos ½αðtÞJ 2 ðzÞ where hLPF1 ðtÞ and hLPF2 ðtÞ are response functions of LPF1 and LPF2. In the third step, we obtain the phase α(t) by solving Eq. (18) under the condition that z¼ 2.45 and G2/G1 ¼J1(2.45)/J2(2.45). After the arctangent calculation using CORDIC, the phase α(t) can be demodulated: αðtÞ ¼ arctan½V 1 ðtÞ=V 2 ðtÞ

ð15Þ

A compensation voltage VC(t) ¼β(t)/kD is generated by DA module, satisfying β(t) þα(t)¼0, where kD is the voltage-phase coefficient. In this way, the phase diversity caused by external disturbances can be eliminated, and a well stability of the fringe

Fig. 7. Four images E–E when deformation is set to 3 mm.

B. En et al. / Optics & Laser Technology 59 (2014) 137–142

phase is achieved. The compensation voltage is given by arctan½V 1 ðtÞ=V 2 ðtÞ V C ðtÞ ¼  kD

ð16Þ

141

iterations. Define F(n) as the whole quantization error, and it is given by " # 4

n1

n1

j¼1

i¼j

FðnÞ 8 Q ½V ðnÞ  VðnÞ ¼ eðnÞ þ ∑

∏ PðiÞeðiÞ

ð19Þ

3.2. CORDIC phase calculation The CORDIC is used for arctangent calculations here, and it works by rotating the input vector through constant angles until the angle is reduced to zero. The angle offsets are selected so that the operations are only shifts, adds/subs and compares. The original work about CORDIC is credited to Ref. [12,13]. Assuming ! that the input vector is V ¼ ðx; yÞ and enable the vectoring mode, the CORDIC can calculate the amplitude and phase with pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 =K n and arctan (y/x), respectively. Kn is the gain coefficient, and n is the final iteration times. However, the quantization in CORDIC will bring error to the phase α(t). In the iterations, define Q ½d as the quantization operation and ‖PðiÞ‖ ¼ K i , where i is the iterative number and 0 r ir n. we obtain Q ½VðiÞ ¼ VðiÞ þ eðiÞ

Combining Eqs. (18) and (19), we obtain  " # n  1 n  1    FðnÞ r jeðnÞj þ jeðiÞj U  ∑ ∏ PðiÞ  j ¼ 1 i ¼ j   ( " #) n  1 n  1  pffiffiffi  b  1 D   r 2U2 1 þ  ∑ ∏ PðiÞ  r pffiffiffi b j ¼ 1 i ¼ j  2U2

ð20Þ

where D ¼ 1 þ j∑nj ¼ 11 Π ni ¼ 1j PðiÞj, and D is converged on a constant. According to Eq. (20), it turns out that the quantization error is mainly depending on b and n. When choosing b ¼32 and n ¼15, the maximum error is about 6.1
10  5 rad, as can be seen graphically in Fig. 4.

ð17Þ

where V ðiÞ ¼ ½xi yi , and eðiÞ ¼ ½ex ðiÞ ey ðiÞ. ex(i) and ey(i) are quantization error for xi and yi, respectively. Assuming xi and yi get b bits of data, it is obvious that ex ðiÞ r 2  b  1 and ey ðiÞ r 2  b  1 . Making modulus operation of e(i) yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð18Þ jeðiÞj 8 e2x þ e2y r 2 U 2  b  1 The quantization error in CORDIC is composed of two parts: the quantization error in the current iteration and in the former all

4. Experiment The experimental setup depicted in Fig. 1 is constructed. A He– Ne laser operated at 633 nm wavelength and a single mode 2
2 fiber-optic coupler are used. The optimum SPM depth z is set to be 2.45. The frequency of the sinusoidal modulation signal for PZT2 is 2 KHz. The center frequency of the BPF1 Fig. 3 was 2 KHz while the BPF2 is 4 KHz. The cutoff frequency of the LPFs is 200 Hz.

Fig. 8. The 3D profile reconstruction of rubber membrane deformation.

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4.1. kD calibration To generate the compensation voltage VC(t), the voltage-phase coefficient kD needs to be calibrated. As shown in Fig. 5, when PZT2 is modulated by a 100 Hz triangular wave, S(t) received by PD is depicted in red line. As the phase change for Michelson interferometer reaches 2π rad, the modulating voltage variation is ΔU¼0.198 V. Considering the PZT driver's amplification coefficient is Adriver ¼ 177:5, the voltagephase coefficient kD is 178.78 mrad/V. 4.2. Phase stability measurement To verify the performance of PSS, we capture S(t) for 1 s with PSS not operating, as shown in Fig. 6(a). S(t) in the region of [100,102.5], [600,604] and [900,904] ms are picked up, and depicted in red line. Comparing the three wave shapes, we know that the amplitude and phase of S(t) drift with a low frequency due to the influence of external disturbances. When the compensation voltage VC(t) is generated for the PZT driver according to Eq. (16), the phase of interference fringes will be well stabilized in about 0.5 ms. The phase shift induced by external disturbances is eliminated as shown in Fig. 6(b). The phase α(t) is demodulated from Eq. (15) and shown in Fig. 6(c). The error of the demodulated phase α(t) is less than 2.75 mrad. From the frequency spectrum of S(t) shown in Fig. 6(d), the amplitude of second harmonic component is superior to 18.1 dB comparing to the first harmonic component. From Eq. (13), the phase error in Michelson interferometer is less than 106.86 mrad, and the phase error for interference fringes is less than 53.43 mrad. 4.3. Dynamic measurement To verify the feasibility of dynamic 3D profile measurement, the experimental setup is depicted in Fig. 1. Besides, a rubber membrane with dynamic deformation drove by mechanical equipments has been fixed on the reference plane. The interference fringes are projected on the rubber membrane. A CCD image sensor (600
600 pixels) operating at 4 KHz, integrates the time-varying intensity on each pixel in parallel during the four quarters of the modulation period successively. When the deformation of rubber membrane is set to 3 mm, four images E1–E4 are shown in Fig. 7. In the same way, when the heights are set to 0.2 mm, 0.5 mm, 1.5 mm, 3.0 mm, the measured results are 0.211 mm, 0.477 mm, 1.520 mm, 2.974 mm. The RMSE (root-mean-square error) is about 0.018 mm. The 3D profile of the rubber membrane has been reconstructed in Fig. 8. For a image of 600
600 pixels, the time of image enhancement and filter is about 45 ms, and the time for the phase extraction is about 128 ms, the profile reconstruction time is about 56 ms. Taking account of other time consumption, the total time is less than 250 ms on the ordinary desktop

computer platform. The dynamic deformation measurement of the rubber membrane can be achieved.

5. Conclusion We have studied a all-fiber SPM system for 3D profilometry. Optimum SPM parameters minimizing the effect of the additive noise have been adjusted in practice. The closed-loop phase stabilization system combining the PGC method and the CORDIC has achieved a precision of 2.75 mrad, and external disturbances are estimated to be of the order of 10  2 rad. By measuring the dynamic deformation of the rubber membrane, RMSE is about 0.018 mm, and a single measurement takes less than 250 ms, which is feasible for real-time application. To improve the measuring accuracy, a new method of dual-wavelength interferometry system for 3D profile measurement is now under research.

Acknowledgment This work was supported by the National High Technology Research and Development Program of China (2013AA102402) and the National Natural Science foundation of China (51275349). References [1] Su XY, Zhang QC. Dynamic 3-D shape measurement method: a review. Opti Lasers Eng 2010;48(2):191–204. [2] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. Opt Soc Am 1982;72(1):156–60. [3] Su XY, Chen WJ, Zhang QC. Dynamic 3-D shape measurement method based on FTP. Opti Lasers Eng 2001;36(1):49–64. [4] Srinivasan V, Liu HC, Halioua M. Automated phase-measuring profilometry of 3-D diffuse objects. Appl Opt 1984;23(18):3105–8. [5] Zhang H, Lalor MJ, Burton DR. Spatio temporal phase unwrapping for the measurement of discontinuous objects in dynamic fringe-projection phaseshifting profilometry. Appl Opt 1999;38(16):3534–41. [6] Meadows DM, Johnson WO, Allen JB. Generation of surface contours by moiré patterns. Appl Opt 1970;9(4):942–7. [7] Sajan MR, Tay CJ, Shang HM. TDI imaging and scanning moiré for online defect detection. Opt Laser Technol 1997;29(6):327–31. [8] Su LK, Su XY, Li WS. Application of modulation measurement profilometry to objects with surface holes. Appl Opt 1999;38(7):1153–8. [9] Dubois A. Phase-map measurements by interferometry with sinusoidal phase modulation and four integrating buckets. Opt Soc Am 2001;18(8):1972–9. [10] Wang B, Wang X, Li Z. Sinusoidal phase-modulating laser diode interferometer insensitive to intensity modulation for real-time displacement measurement with feedback control system. Opt Commun 2012;285(18):3827–31. [11] Wang GQ, Xu TW, Li F. PGC demodulation technique with high stability and low harmonic distortion. IEEE Phon Technol Lett 2012;24(23):2093–6. [12] Volder JE. The CORDIC trigonometric computing technique. IRE Trans Electron Comput 1959;3:330–4. [13] Walther JS., A unified algorithm for elementary functions. In: Spring joint computer conference; 1971. p. 379–85.