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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 2845–2852
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Heterodyne moiré surface profilometry

Wei-Yao Chang, Fan-Hsi Hsu, Kun-Huang Chen, Jing-Heng Chen, and Ken Y. Hsu  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 2845-2852 (2014)
http://dx.doi.org/10.1364/OE.22.002845


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Abstract

In this study, a novel moiré fringe analysis technique is proposed for measuring the surface profile of an object. After applying a relative displacement between two gratings at a constant velocity, every pixel of CMOS camera can capture a heterodyne moiré signal. The precise phase distribution of the moiré fringes can be extracted using a one-dimensional fast Fourier transform (FFT) analysis on every pixel, simultaneously filtering the harmonic noise of the moiré fringes. Finally, the surface profile of the tested objected can be generated by substituting the phase distribution into the relevant equation. The findings demonstrate the feasibility of this measuring method, and the measurement error was approximately 4.3 μm. The proposed method exhibits the merits of the Talbot effect, projection moiré method, FFT analysis, and heterodyne interferometry.

© 2014 Optical Society of America

1. Introduction

2. Principle

Figure 1
Fig. 1 Optical configuration. MO: microscopic objective; PH: pinhole; L1: collimating lens; L2: imaging lens; L3: camera lens; G1 and G2: linear gratings; M1 and M2: translation motorized stage; C: CMOS camera; Z1: first Talbot distance; F: focal length of imaging lens; α: projection angle; S: sample.
shows the optical configuration of the proposed approach. For convenience, the z-axis is assigned as the observation axis of the CMOS camera C, and the y-axis is set perpendicular to the paper plane. A laser light beam (wavelength λ) passes through a beam expander to form an expanded and collimated light beam. Subsequently, the light beam impinges on linear grating G1 at projection angle α forming a self-image of grating G1 and projecting the first self-image on the test surface. The first self-image distance Z1 can be expressed as follows [19

19. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129(3-4), 167–172 (1996). [CrossRef]

]:
Z1=p2λcos3α,
(1)
where p is the pitch of grating G1. The projected fringes are deformed because of the surface profile; this can be expressed as follows:
q1(x,y)=12{1+γcos[2πp(xz(x,y)tan(α))ϕ1]},
(2)
where γ is the visibility of the deformed fringes, z(x, y) is the height distribution of the tested surface, and ϕ1 is the initial phase of grating G1. The deformed fringes are imaged at 1 × magnification on reference grating G2 with a grating pitch of p to form the moiré fringes; the image was captured using a CMOS camera C. The captured fringes can be expressed as follows:
q2(x,y)=q1(x,y)×W(x,y)=12{1+γcos[2πp(x+z(x,y)tanα)+ϕ1]}×12[1+cos(2πpx+ϕ2)],
(3)
where q1(−x, −y) denotes the imaging fringes of q1(x, y) by using the imaging lens L2; W(x, y) is the transmission function of grating G2; and ϕ2 is the initial phase of grating G2. Equation (3) can be written as follows:
q2(x,y)=14{1+γcos[2πp(x+z(x,y)tanα)+ϕ1]+cos(2πpx+ϕ2)+γ2cos[2πp(2x+z(x,y)tanα)+ϕ1+ϕ2]+γ2cos[ϕ(x,y)+ϕ1ϕ2]},
(4)
and
ϕ(x,y)=2πtanαpz(x,y),
(5)
where ϕ(x, y) is the phase distribution of the moiré fringes. In Eq. (4), the second, third, and fourth terms are the harmonic noise, and the last term is the moiré fringe, which is primarily relative to the height distribution of the tested surface. To obtain the phase distribution ϕ, gratings G1 and G2 are moved at a constant speed, v along the −x-direction by using the motorized translation stages M1 and M2, respectively. Thus, every pixel of the CMOS camera C can receive the time-varying signal and the moiré fringes can be expressed as follows:
q3(t,x,y)=q1(t,x,y)×W(t,x,y)=12{1+γcos[2πf0t+2πxp+ϕ+ϕ1]}×12[1+cos(2π(f0)t+2πxp+ϕ2)],
(6)
where f0 = v/p is the frequency introduced by the grating movement. Equation (6) can be rewritten as:
q3(t,x,y)=14{1+γcos(2πf0t+2πxp+ϕ+ϕ1)+cos[2π(f0)t+2πxp+ϕ2]+γ2cos(4πxp+ϕ+ϕ1+ϕ2)+γ2cos(2πfht+ϕ+ϕ1ϕ2)},
(7)
where the last term is not relative to the grating line along the x-direction and fh = 2f0 is the heterodyne moiré frequency. For convenience, Eq. (7) can be rewritten using the Euler formula as follows [10

10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

]:
q3(t,x,y)=18{[a(t,x,y)+a*(t,x,y)]+[b(t,x,y)exp(i2πf0t)+b*(t,x,y)exp(i2πf0t)]+[c(t,x,y)exp[i2π(f0)t]+c*(t,x,y)exp[i2π(f0)t]]+12[d(t,x,y)exp(i2πfht)+d*(t,x,y)exp(i2πfht)]},
(8)
where * denotes the complex conjugate operator, and
a(t,x,y)=1+γ2exp[i(4πxp)+ϕ+ϕ1+ϕ2],
(9)
b(t,x,y)=γexp[i(2πxp+ϕ+ϕ1)],
(10)
c(t,x,y)=γ2exp[i(2πxp+ϕ2)],and
(11)
d(t,x,y)=γ2exp[i(ϕ+ϕ1ϕ2)].
(12)
Applying the FFT to Eq. (8), Eq. (13) yields the following:
Q(f,x,y)=18{A(f,x,y)+A*(f,x,y)+B(ff0,x,y)+B*(f+f0,x,y)+12[C(f+f0,x,y)+C*(ff0,x,y)]+12[D(ffh,x,y)+D*(f+fh,x,y)]},
(13)
where Q, A, B, C, and D are the Fourier transform of q3, a, b, c, and d. Subsequently, D(ffh, x, y) is shifted to D(f, x, y) and processed using the inverse Fourier transform to obtain d(t, x, y). Calculating a logarithm of d(t, x, y) yields:
ln[d(x,y)]=ln(γ2)+i(ϕ+ϕ1ϕ2).
(14)
Extracting the imaginary part of Eq. (14) allows generating ϕ + ϕ1ϕ2. To generate phase distribution ϕ(x, y), this procedure was applied to every pixel of camera C, and the initial phase difference ϕ1ϕ2 was neglected because it did not influence the relative height distribution of the surface measurement. Considering the image reversal that results from the imaging lens, Eq. (5) can be rewritten as follows:
z(x,y)=p2πtanαϕ(x,y).
(15)
Finally, substituting ϕ(x, y) into Eq. (15) and considering the image reversal that resulted from the imaging lens, the surface profile of the tested object can be reconstructed.

3. Experimental results and discussions

To validate the feasibility of the proposed method, a coin was measured [Fig. 2(a)
Fig. 2 Sample image and experimental results. (a) coin image; (b) moiré fringes on coin surface; (c) phase map of moiré fringes on coin surface.; (d) reconstructed coin surface profile.
]. The experimental setup comprised a diode laser (wavelength = 473 nm), two linear gratings (pitch = 0.2822 mm), an imaging lens with a focal length of 200 mm, two motorized translation stages (Sigma Koki/SGSP(MS)26 −100) with resolution of 0.05 μm to generate a heterodyne frequency fh = 20 Hz (v = 0.2822 mm/s), and a CMOS camera (Basler /A504k) with an 8-bit gray level and 1280 × 1024 image resolution. For convenience, the projection angle was set at 30°, the frame rate of the CMOS camera was set at fs = 120 fps, the exposure time was set at te = 8 ms, and the total recording time was set at T = 1 s to record the heterodyne moiré signals at various times. Figures 2(b)-2(d) display the experimental results. Figure 2(b) is a recorded moiré image, showing that grating noise spread throughout the image. Despite the nonuniform light intensity distribution, nonuniform scattering property of the coin surface, and the height distribution of the coin, which is lower than that one period of the moiré fringe can represent, the phase distribution of the moiré fringes were extracted by using the proposed fringe analysis method [Fig. 2(c)]. The surface profile of the coin was reconstructed using Eq. (15), as shown in Fig. 2(d). To prove the accuracy of the proposed method, the contour from A to A′ [Fig. 2(a)] was measured using a stylus profilometer (KLA-Tencor/Alpha Step D-100) toserve as reference data. Figure 3
Fig. 3 Contour lines depicted from experimental and reference data.
shows the contour lines from the experimental and reference data, indicating that the experimental and reference data exhibited the same height trends; this proves the feasibility of the proposed method.

According to Eq. (6), the height error Δz yielded by using the proposed method can be expressed as follows:
Δz=|zpΔp|+|zαΔα|+|zϕΔϕ|,
(16)
where Δp is the grating pitch error, Δα is the projection angle error, and Δϕ is the phase error. The grating pitch error resulted from fabrication error. Because the gratings in the experiment were produced by plating chromium on a glass plate using a mask laser writer, the grating pitch error Δp was approximately 0.1 μm. The projection angle error resulted from axis alignment error and the resolution of the rotary stage. When the axis alignment error was 0.05° and the resolution of the rotary stage was 10 arcmin, the projection angle error Δα was approximately 0.22°. The phase error of the proposed method primarily resulted from the sampling error. When heterodyne moiré signals are captured using a CMOS camera, using an exposure and integration process, the recorded intensity of one pixel at the kth sampling point can be expressed as follows:
qc(kts)=1tektskts+teq3dt=14{1+γsinc(f0te)cos(2πf0kts+2πxp+ϕ+ϕ1+πf0te)+sinc(f0te)cos[2π(f0)kts+2πxp+ϕ2+πf0te]+γ2cos(4πxp+ϕ+ϕ1+ϕ2)+γ2sinc(fhte)cos[2π(fh)kts+ϕ+ϕ1ϕ2+πfhte]},
(17)
where ts = 1/fs is the frame period. The intensity is subsequently quantized in gray-level units as:
qd(kts)=round(qc×2n),
(18)
where n denotes the number of the gray level. After substituting the experimental conditions in Eqs. (17) and (18), the captured heterodyne moiré signal of one pixel can be simulated and analyzed using FFT. The phase error Δϕ can be obtained by comparing the calculated phases of FFT and set ϕ when the relative phase measurement of the surface profilometry is considered and the light intensity error resulting from the stability of the light source and grating vibration is 1%. The phase error Δϕ of the proposed method can be estimated as approximately 1.53°. Substituting the experimental conditions, Δp, Δα, and Δϕ into Eq. (16) yielded a height error Δz of approximately 4.3 μm.

The visibility of the projected fringes influences the amplitude of heterodyne moiré signals, and affects the recorded magnitude of heterodyne frequency component, namely d(x, y) in Eq. (12). It is obvious that if visibility is zero, the heterodyne frequency component and the phase of this component certainly cannot be extracted by FFT. Therefore, the suggested minimum visibility is about 0.4 because the height error in this condition reaches 6.67 μm. The lower visibility causes much larger height error and results in a disappointing result. Besides, the angular misalignment between the directions of the projected fringes and the grating line of G2 can reach 0.01 arcmin in our experiment. The resultant phase variation caused by this small angle in whole image can be estimated merely within 1.7 × 10−5 degree [4

4. J. Dhanotia and S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. 50(10), 1446–1452 (2011). [CrossRef] [PubMed]

], hence the resultant small angular misalignment can be neglected.

The proposed heterodyne moiré interferometry is analogous to the conventional heterodyne interferometry [20

20. D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. 27(1), 19–23 (1996). [CrossRef]

]. The heterodyne moiré frequency is introduced by the beat frequency between two grating movement frequencies, and analyzing the heterodyne moiré signal can achieve the measurement purpose. In addition, according to the principle, the experimental results, and the error analysis, the proposed method can avoid losing the detail information of moiré fringes because the grating noise is filtered out pixel by pixel. Hence, this approach has higher accuracy than the techniques that filter out the grating noise in spatial frequency domain, such as conventional Fourier transform method whose maximum detectable spatial frequency on the surface is limited by the spatial frequency of the carrier fringes or the grating noise [10

10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

], and some phase shifting methods which filter the moiré images in every phase step. Besides, phase shifting method combined with the optical compensation method is a powerful technique to filter out the grating noise of moiré and to extract the accurate phase data. However, this approach can just be implemented by continuously moving two gratings in a time interval and recording continuously time-varying signals. Analyzing the captured signals can filter out the grating noise and extract the phase data at the same time. Therefore, the proposed method can avoid the probable errors that may be introduced when controlling and matching the displacement of motorized translation stage and the exposure time of camera in the optical compensation. Furthermore, the recording time of this method is less than the optical-compensation phase shifting method in the same conditions of the grating speed and the frame rate of camera. Hence, the proposed method has higher stability due to the less measurement time. It is worthy to mention that applying spatial light modulator or LCD projector to moiré profilometry has been developed to improve the instability caused from the mechanical vibration [16

16. J. A. N. Buytaert and J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express 16(1), 179–193 (2008). [CrossRef] [PubMed]

, 17

17. J. J. J. Dirckx, J. A. N. Buytaert, and S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. 48(2), 244–250 (2010). [CrossRef]

].

4. Conclusion

A novel moiré fringe analysis technique is proposed for measuring the surface profile of an object, integrating the Talbot effect, projection moiré method, FFT analysis, and heterodyne interferometry. To validate the proposed approach, a coin was measured, yielding a height error of approximately 4.3 μm. Because the grating noise is filtered out pixel by pixel, the proposed method can avoid losing the detail information of moiré fringes. This approach can just analyze the recorded continuously time-varying signals to simultaneously filter out the grating noise and to extract the phase data, hence it can reduce the probable errors resulted from controlling and matching procedures. Additionally, this method demonstrates the advantages of the combined techniques, yielding high precision, high resolution, high stability, and applicability in a large measurement range.

Acknowledgments

The authors would like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 101-2221-E-009-112-MY3 and NSC 102-2221-E-035-057.

References and links

1.

H. Takasaki, “Moiré topography from its birth to practical application,” Opt. Lasers Eng. 3(1), 3–14 (1982). [CrossRef]

2.

J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, “Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms,” Appl. Opt. 27(6), 1164–1169 (1988). [CrossRef] [PubMed]

3.

A. A. Mudassar and S. Butt, “Self-imaging-based laser collimation testing technique,” Appl. Opt. 49(31), 6057–6062 (2010). [CrossRef]

4.

J. Dhanotia and S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. 50(10), 1446–1452 (2011). [CrossRef] [PubMed]

5.

J. Y. Lee, Y. H. Wang, L. J. Lai, Y. J. Lin, and Y. H. Chang, “Development of an auto-focus system based on the moiré method,” Measurement 44(10), 1793–1800 (2011). [CrossRef]

6.

Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]

7.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform,” Opt. Commun. 136(5-6), 370–374 (1997). [CrossRef]

8.

M. Ramulu, P. Labossiere, and T. Greenwell, “Elastic–plastic stress/strain response of friction stir-welded titanium butt joints using moiré interferometry,” Opt. Lasers Eng. 48(3), 385–392 (2010). [CrossRef]

9.

K. S. Lee, C. J. Tang, H. C. Chen, and C. C. Lee, “Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method,” Appl. Opt. 47(13), C315–C318 (2008). [CrossRef] [PubMed]

10.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

11.

Y. M. He, C. J. Tay, and H. M. Shang, “Deformation and profile measurement using the digital projection grating method,” Opt. Lasers Eng. 30(5), 367–377 (1998). [CrossRef]

12.

S. Mirza and C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng. 44(1), 013601 (2005). [CrossRef]

13.

C. Quan, Y. Fu, and C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230(1-3), 23–33 (2004). [CrossRef]

14.

Y. B. Choi and S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998). [CrossRef]

15.

J. A. N. Buytaert and J. J. J. Dirckx, “Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation,” J. Opt. Soc. Am. A 24(7), 2003–2013 (2007). [CrossRef] [PubMed]

16.

J. A. N. Buytaert and J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express 16(1), 179–193 (2008). [CrossRef] [PubMed]

17.

J. J. J. Dirckx, J. A. N. Buytaert, and S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. 48(2), 244–250 (2010). [CrossRef]

18.

J. A. N. Buytaert and J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011). [CrossRef]

19.

M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129(3-4), 167–172 (1996). [CrossRef]

20.

D. C. Su, M. H. Chiu, and C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. 27(1), 19–23 (1996). [CrossRef]

OCIS Codes
(040.2840) Detectors : Heterodyne
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects
(120.4120) Instrumentation, measurement, and metrology : Moire' techniques
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: October 24, 2013
Revised Manuscript: January 20, 2014
Manuscript Accepted: January 29, 2014
Published: January 31, 2014

Citation
Wei-Yao Chang, Fan-Hsi Hsu, Kun-Huang Chen, Jing-Heng Chen, and Ken Y. Hsu, "Heterodyne moiré surface profilometry," Opt. Express 22, 2845-2852 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2845


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References

  1. H. Takasaki, “Moiré topography from its birth to practical application,” Opt. Lasers Eng. 3(1), 3–14 (1982). [CrossRef]
  2. J. J. J. Dirckx, W. F. Decraemer, G. Dielis, “Phase shift method based on object translation for full field automatic 3-D surface reconstruction from moire topograms,” Appl. Opt. 27(6), 1164–1169 (1988). [CrossRef] [PubMed]
  3. A. A. Mudassar, S. Butt, “Self-imaging-based laser collimation testing technique,” Appl. Opt. 49(31), 6057–6062 (2010). [CrossRef]
  4. J. Dhanotia, S. Prakash, “Automated collimation testing by incorporating the Fourier transform method in Talbot interferometry,” Appl. Opt. 50(10), 1446–1452 (2011). [CrossRef] [PubMed]
  5. J. Y. Lee, Y. H. Wang, L. J. Lai, Y. J. Lin, Y. H. Chang, “Development of an auto-focus system based on the moiré method,” Measurement 44(10), 1793–1800 (2011). [CrossRef]
  6. Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162–3166 (1985). [CrossRef] [PubMed]
  7. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using a digital Fourier transform,” Opt. Commun. 136(5-6), 370–374 (1997). [CrossRef]
  8. M. Ramulu, P. Labossiere, T. Greenwell, “Elastic–plastic stress/strain response of friction stir-welded titanium butt joints using moiré interferometry,” Opt. Lasers Eng. 48(3), 385–392 (2010). [CrossRef]
  9. K. S. Lee, C. J. Tang, H. C. Chen, C. C. Lee, “Measurement of stress in aluminum film coated on a flexible substrate by the shadow moiré method,” Appl. Opt. 47(13), C315–C318 (2008). [CrossRef] [PubMed]
  10. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
  11. Y. M. He, C. J. Tay, H. M. Shang, “Deformation and profile measurement using the digital projection grating method,” Opt. Lasers Eng. 30(5), 367–377 (1998). [CrossRef]
  12. S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng. 44(1), 013601 (2005). [CrossRef]
  13. C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230(1-3), 23–33 (2004). [CrossRef]
  14. Y. B. Choi, S. W. Kim, “Phase-shifting grating projection moiré topography,” Opt. Eng. 37(3), 1005–1010 (1998). [CrossRef]
  15. J. A. N. Buytaert, J. J. J. Dirckx, “Design considerations in projection phase-shift moiré topography based on theoretical analysis of fringe formation,” J. Opt. Soc. Am. A 24(7), 2003–2013 (2007). [CrossRef] [PubMed]
  16. J. A. N. Buytaert, J. J. J. Dirckx, “Moiré profilometry using liquid crystals for projection and demodulation,” Opt. Express 16(1), 179–193 (2008). [CrossRef] [PubMed]
  17. J. J. J. Dirckx, J. A. N. Buytaert, S. A. M. Van Der Jeught, “Implementation of phase-shifting moire profilometry on a low-cost commercial data projector,” Opt. Lasers Eng. 48(2), 244–250 (2010). [CrossRef]
  18. J. A. N. Buytaert, J. J. J. Dirckx, “Study of the performance of 84 phase-shifting algorithms for interferometry,” J. Opt. 40(3), 114–131 (2011). [CrossRef]
  19. M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129(3-4), 167–172 (1996). [CrossRef]
  20. D. C. Su, M. H. Chiu, C. D. Chen, “A heterodyne interferometer using an electro-optic modulator for measuring small displacement,” J. Opt. 27(1), 19–23 (1996). [CrossRef]

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