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An orthogonal phase-shifting interferometry and its application to the measurement of optical plate
Optik 127 (2016) 8841–8846
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
An orthogonal phase-shifting interferometry and its application to the measurement of optical plate Yingming Zhao a,∗ , Ruofu Yang a,b , Chunping Yang a a
School of Opto-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, PR China Integrated Optoelectronic Technology Center, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, PR China b
a r t i c l e
i n f o
Article history: Received 30 August 2015 Received in revised form 9 June 2016 Accepted 28 June 2016 Keywords: Phase-shifting interferometry Optical testing Phase measurement Jones matrix PZT phase shifter
a b s t r a c t An optical interferometry named orthogonal phase-shifting interferometry (OPSI) was proposed. Compared with other interferometers, OPSI works well at air turbulence and complex environment. Phase recovery method was calculated by Jones matrix for measurement of optical elements surface shape and roughness. In order to control piezo-electric transducer (PZT) shifter, the property between voltage and phase of PZT shifter was calculated. Based on this interferometry, the roughness of an unknown transparent optical plate was measured. The results show that the error of peak to valley (PV) value is 0.0087 ( = 1064 nm) and the error of root mean square (RMS) value is 0.0006 compared with WYKO(a commercial interferometer of Veeco),and this demonstrates the high stability and precision of the orthogonal phase-shifting interferometry. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Optical interferometry has important significance in optical testing and the field of measurement technology. It is used for measurement of surface shape and roughness, and it has lots of advantages such as non-contact and fast response [1–5]. In the early days of interferometry, fringe patterns were analyzed manually. Recently, the development of image processing systems allows for automatic analysis of interferograms in optical methods of measurement. Phase shifter is applied by changing the optical path of reference beam [6–8]. One mirror is fastened on a piezo-electric transducer (PZT) shifter to realize the demanded shift movement. But current interferometers with PZT shifter are unstable working at air turbulence and complex environment. The precision and accuracy of measurement are affected. In order to solve above problems, an optical interferometry named orthogonal phase-shifting interferometry (OPSI) was proposed. The system is improved on common path structure. That means that it keeps stable working at complex environment. The property between voltage and phase of PZT shifter was calculated, so the phase shifter can be control well and easily to be operated. Based on this interferometry, the roughness of an unknown transparent optical plate was measured. The accuracy and precision of measurement is proved by comparing its result with the result of WYKO (a commercial interferometer of Veeco).
∗ Corresponding author at: University of Electronic Science and Technology of China, School of Opto-electronic Information, No. 4, Section 2, North Jianshe Road, Sichuan Province, 610054 Chengdu, PR China. E-mail addresses: [email protected] (Y. Zhao), yang [email protected] (R. Yang), [email protected] (C. Yang). http://dx.doi.org/10.1016/j.ijleo.2016.06.096 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Y. Zhao et al. / Optik 127 (2016) 8841–8846
Fig. 1. Schema of the orthogonal phase-shifting interferometry; beam expander (BE), linear polarizer (P1, P2), half wave plate (HWP), polarizing beam splitter (PBS), quarter wave plate (QWP1, QWP2), mirror (M1,M2), piezo transducer (PZT).
2. Principle 2.1. Principle of OPSI measurement As is shown in Fig. 1, the Nd: YAG laser beam (1064 nm, 50 mW) transmits through the beam expander (BE) and reaches the P1 (polarizer) formed linear polarized light. Linear polarized light transmits through the half wave plate (HWP) following the X-axis direction. Next, linear polarized laser beam is splitted into s and p polarized beams at the beam splitting plane of polarizing beam splitter (PBS).The fast axis of the quarter wave plate (QWP) is rotated by 45◦ . After the passage of the signal wave through QWP1, linear polarized laser beam becomes circular polarization. And it carries the information of the unknown optical plate. Mirrors M1, M2 placed equidistant from PBS, and M2 is fastened on a PZT shifter thus realizing the demanded phase shift movement. The s polarized beam split by PBS passes QWP2 and reaches M2, and it is the polarized reference wave. Finally, these two waves (the signal one and the reference one) pass through P2 (polarizer) and constitute the orthogonally polarized beams in common path. Two beams interfere an intensity fringe on the CCD camera.
2.2. Phase recovery method In the case of a simplified model of the OPSI system, only the optical device under test is considered as a phase modulation. Jones matrix is commonly applied to the field of polarization interferometry [9]. Jones vector of the incident beam is E1 = A1 , and p polarized beam has none phase modulation in the CCD camera image space, while s polarized beam has a B1
phase modulation caused by the unknown optical plate. The Jones matrix of s polarized beam is
through optical elements in path, the Jones vector reached CCD camera is E1 =
E1 =
7
J8−i E1
i=1
1 0 , After transmission 0 0
A1 , and it is simply expressed as: B1
(1)
Y. Zhao et al. / Optik 127 (2016) 8841–8846
8843
where J7 , J6 · · ·J1 are given by
⎡
⎤
⎢ J7 = ⎣
1 sin 2 2 ⎥
cos2
0
0
⎦ ; J6 = 1 0 1 sin 2 sin2 2 1 −i 1 0 1 ; J1 = J2 = √ 2 −i 1 0 0
ei ; J5
0
0
0
1
=
1 ; J4 = √ 2
1
−i
−i
1
−1
0
0
−1
; J3 =
;
J7 , J6 · · ·J1 are the Jones matrices of polarized optical elements in the passage of s polarized beam, and ϕ is phase and modulation caused by the unknown optical plate. is the angle between transmission axis of P2(polarizer) and X-axis. The Jones matrix is calculated as:
E1 =
⎡
7
J8−i E1 = ⎣
⎤
A1 i sin 2 2 ⎦ ei
(2)
2
A1 isin
i=1
The matrix of p polarized beam is
0 0 . After passing optical elements,the matrix becomesE2 = 0 1
A2 . and it is simply B2
expressed as:
E2 =
7
(3)
J8−i E1
i=1
where J7 , J6 · · ·J1 are given by
⎡ ⎢ J7 = ⎣
⎤
1 sin 2 2 ⎥
cos2
1 0
⎦ ; J6 =
1 0 sin2 sin 2 2 1 −i 0 0 1 J2 = √ ; J1 = 2 −i 1 0 1
1 0
; J5 = 0
0
0
1 ; J4 = √ 2
1
−i
−1
0
; J3 = −i
1
2 i
e 0
(2)
;
−1
J7 , J6 · · ·J1 are the Jones matrices of polarized optical elements in the passage of p polarized beam, where is phase shift caused by PZT shifter with M2. The matrix is calculated as:
⎡
B1 icos2
⎤
2 (2) i ⎦e E2 = ⎣ B i 1 sin 2 2
(4)
After recombining the two beams, the formula is given by:
⎡
⎤ 2 (2) ⎢ A1 sin 2ei + 2B1 cos2 e ⎥ i
E = E1 + E2 =
i 2
⎢ ⎣
2 (2) i 2A1 sin2 ei + B1 sin 2e
⎥ ⎦
(5)
In general, is equal to 45◦ , so the intensity fringe at the CCD camera is calculated as: I = |E | = 2
1 1 2 4 A1 + B12 + A1 B1 cos − 4 2
(6)
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Y. Zhao et al. / Optik 127 (2016) 8841–8846
Fig. 2. Interferograms under different PZT shifter driving voltages. (a)0 V, (b)0.5 V, (c)1.0 V, (d)1.5 V, (e)2.0 V, (f)2.5 V.
If values are 0, /8, /4, 3/8, the intensity fringes are expressed as: I1 =
1 1 2 A1 + B12 + A1 B1 cos 4 2
I2 =
1 1 2 A1 + B12 + A1 B1 sin 4 2
I3 =
1 1 2 A1 + B12 − A1 B1 cos 4 2
I1 =
1 1 2 A1 + B12 − A1 B1 sin 4 2
(7)
ϕ can be quantitatively reconstructed using the formula: ϕ = tan−1
I4 − I2 I3 − I1
(8)
3. Experiment A computer program is employed to control PZT shifter driving controller and CCD camera. The PZT drive voltage step is 0.01 V, and 14 V is the upper boundary. 1400 pieces 600 × 600 pixel interferograms were captured by CCD camera within the range of 0–14 V. Fig. 2 shows interferograms under different PZT shifter driving voltages. The central fringe of each interferogram is marked with a rectangular box. When the voltage is increased to 2.5 V, the marked area has been appeared right horizontal movement. The rectangular box moves from 200× axis pix position to near 400× axis pix position. This indicates that the total interference fringes have displacement caused by the PZT shifter. Throughout the OPSI system, the phase shifting is introduced by the PZT shifter, so the relationship between PZT driving voltage and the phase variety reflects the characteristics of the PZT. Interferograms of each frame are processed by antiGaussian filter and median filter [10–12], and a sine fitting algorithm is used. The initial phases of 1400 pieces interferograms were calculated. The property curve between voltage and phase of PZT shifter is drawn in Fig. 3. In order to verify the testing result of OPSI, an unknown optical plate is tested by OPSI and a common commercial interferometer WYKO [11–14]. The result measured by WYKO is presented as a standard to analyze the error and accuracy of OPSI. The OPSI structure of the experimental setup is shown in Fig. 4. The PZT driving voltages required at Eq. (7) can be got from Fig. 3, and they are 0 V, 2.027 V, 3.380 V, 5.465 V realizing 0, /8, /4, 3/8 phase shift. After capturing 4 pieces interferograms at 0 V, 2.027 V, 3.380 V, 5.465 V,the wrapped phase can be calculated by Eq. (8).The unwrapped phase of the unknown optical plate is obtained by least squares method shown in Fig. 5(b). Fig. 5 shows the results measured by WYKO and OPSI using the unknown optical plate. The phase distribution of the optical plate shown in Fig. 5(a) is similar to Fig. 5(b).This illustrates the accuracy of testing results by OPSI. Errors are common in practice for various reasons including
Y. Zhao et al. / Optik 127 (2016) 8841–8846
8845
Fig. 3. PZT driving voltage and phase shifting curve.
Fig. 4. OPSI experimental setup.
Fig. 5. (a) The phase distribution of optical plate measured by WYKO (b)the phase distribution of optical plate measured by OPSI.
nonlinear and irregular ones [13]. Table 1 is given the detail data measured by OPSI and WYKO, and the error of PV value and RMS value is calculated.
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Y. Zhao et al. / Optik 127 (2016) 8841–8846
Table 1 OPSI and WYKO interferometer testing data comparison.
PV value/ RMS value/ PV value error/ RMS value error/
WYKO interferometer
OPSI
0.0342 0.0026 0.0087 0.0006
0.0255 0.0032
4. Conclusion To summarize, an optical interferometry named OPSI(orthogonal phase-shifting interferometry) with common path structure was proposed. And a reconstructed phase method was calculated by Jones matrix. The results of the optical plate experiment show that the OPSI system has high accuracy compared with WYKO, and it can stay good performance at complex environment. It proves a conceptual correctness and feasibility of the proposed method and its applicability to quantitative high-resolution phase measurements. Acknowledgements This work is supported by National Natural Science Foundation of China (NSFC) under Contract No.61308062. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R. Windecker, H.J. Tiziani, Optical roughness measurements using extended white-light interferometry, Opt. Eng. 38 (1999) 1081–1087. S. Han, E. Novak, J. Sullivan, Application of grazing incidence interferometer to rough surface measurements, Proc. SPIE 5144 (2003) 391–400. M.H. Mahdieh, T. Nazari, Measurement of impurity and temperature variations in water by interferometry technique, Optik 125 (2013) 4393–4396. Y.P. Kumar, S. Chatterjee, Thickness measurement of transparent glass plate using a lateral shearing cyclic path optical configuration setup and polarization phase shifting interferometry, Appl. Opt. 49 (2010) 6552–6557. C. Poilâne, P. Sandoz, Thickness measurement of nontransparent free films by double-side white-lightinterferometry: calibration and experiments, Rev. Sci. Instrum. 77 (2006) 055102. R. Bouchal, R. Celechovsky, Z. Bouchal, Polarization sensitive phase-shifting Mirau interferometry using a liquid crystal variable retarder, Opt. Lett. 40 (2015) 4567–4570. L. Zhuo, X.X. Lu, An effective method in calibrating nonlinear errors for phase-shifting interferometry, Optik 125 (2014) 2109–2112. T. Guo, F. Li, J.P. Chen, X. Fu, X.T. Hu, Multi-wavelength phase-shifting interferometry for micro-structures measurement based on color image processing in white light interference, Opt. Lasers Eng. 82 (2016) 41–47. E. Massa, G. Mana, J. Krempel, M. Jentschel, Polarization delivery in heterodyne interferometry, Opt. Express 21 (2013) 27119–27126. J. Militky, M. Kadulova, P. Hlubina, Highly sensitive displacement measurement based on spectral interferometry and Vernier effect, Opt. Commun. 366 (2016) 335–339. Y.P. Kumar, S. Chatterjee, S.S. Negi, Small roll angle measurement using lateral shearing cyclic path polarization interferometry, Appl. Opt. 55 (2016) 979–983. A. Sanchez-Castillo, S. Eslami, F. Giesselmann, P. Fischer, Circular polarization interferometry: circularly polarized modes of cholesteric liquid crystals, Opt. Express 22 (2014) 31227–31236. Z. Sarac, R. Grob, C. Richter, B. Wiesner, G. Hausler, Optimization of white light interferometry on rough surfaces based on error analysis, Optik 115 (2004) 351–357.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
An orthogonal phase-shifting interferometry and its application to the measurement of optical plate Yingming Zhao a,∗ , Ruofu Yang a,b , Chunping Yang a a
School of Opto-Electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, PR China Integrated Optoelectronic Technology Center, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, PR China b
a r t i c l e
i n f o
Article history: Received 30 August 2015 Received in revised form 9 June 2016 Accepted 28 June 2016 Keywords: Phase-shifting interferometry Optical testing Phase measurement Jones matrix PZT phase shifter
a b s t r a c t An optical interferometry named orthogonal phase-shifting interferometry (OPSI) was proposed. Compared with other interferometers, OPSI works well at air turbulence and complex environment. Phase recovery method was calculated by Jones matrix for measurement of optical elements surface shape and roughness. In order to control piezo-electric transducer (PZT) shifter, the property between voltage and phase of PZT shifter was calculated. Based on this interferometry, the roughness of an unknown transparent optical plate was measured. The results show that the error of peak to valley (PV) value is 0.0087 ( = 1064 nm) and the error of root mean square (RMS) value is 0.0006 compared with WYKO(a commercial interferometer of Veeco),and this demonstrates the high stability and precision of the orthogonal phase-shifting interferometry. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Optical interferometry has important significance in optical testing and the field of measurement technology. It is used for measurement of surface shape and roughness, and it has lots of advantages such as non-contact and fast response [1–5]. In the early days of interferometry, fringe patterns were analyzed manually. Recently, the development of image processing systems allows for automatic analysis of interferograms in optical methods of measurement. Phase shifter is applied by changing the optical path of reference beam [6–8]. One mirror is fastened on a piezo-electric transducer (PZT) shifter to realize the demanded shift movement. But current interferometers with PZT shifter are unstable working at air turbulence and complex environment. The precision and accuracy of measurement are affected. In order to solve above problems, an optical interferometry named orthogonal phase-shifting interferometry (OPSI) was proposed. The system is improved on common path structure. That means that it keeps stable working at complex environment. The property between voltage and phase of PZT shifter was calculated, so the phase shifter can be control well and easily to be operated. Based on this interferometry, the roughness of an unknown transparent optical plate was measured. The accuracy and precision of measurement is proved by comparing its result with the result of WYKO (a commercial interferometer of Veeco).
∗ Corresponding author at: University of Electronic Science and Technology of China, School of Opto-electronic Information, No. 4, Section 2, North Jianshe Road, Sichuan Province, 610054 Chengdu, PR China. E-mail addresses: [email protected] (Y. Zhao), yang [email protected] (R. Yang), [email protected] (C. Yang). http://dx.doi.org/10.1016/j.ijleo.2016.06.096 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
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Y. Zhao et al. / Optik 127 (2016) 8841–8846
Fig. 1. Schema of the orthogonal phase-shifting interferometry; beam expander (BE), linear polarizer (P1, P2), half wave plate (HWP), polarizing beam splitter (PBS), quarter wave plate (QWP1, QWP2), mirror (M1,M2), piezo transducer (PZT).
2. Principle 2.1. Principle of OPSI measurement As is shown in Fig. 1, the Nd: YAG laser beam (1064 nm, 50 mW) transmits through the beam expander (BE) and reaches the P1 (polarizer) formed linear polarized light. Linear polarized light transmits through the half wave plate (HWP) following the X-axis direction. Next, linear polarized laser beam is splitted into s and p polarized beams at the beam splitting plane of polarizing beam splitter (PBS).The fast axis of the quarter wave plate (QWP) is rotated by 45◦ . After the passage of the signal wave through QWP1, linear polarized laser beam becomes circular polarization. And it carries the information of the unknown optical plate. Mirrors M1, M2 placed equidistant from PBS, and M2 is fastened on a PZT shifter thus realizing the demanded phase shift movement. The s polarized beam split by PBS passes QWP2 and reaches M2, and it is the polarized reference wave. Finally, these two waves (the signal one and the reference one) pass through P2 (polarizer) and constitute the orthogonally polarized beams in common path. Two beams interfere an intensity fringe on the CCD camera.
2.2. Phase recovery method In the case of a simplified model of the OPSI system, only the optical device under test is considered as a phase modulation. Jones matrix is commonly applied to the field of polarization interferometry [9]. Jones vector of the incident beam is E1 = A1 , and p polarized beam has none phase modulation in the CCD camera image space, while s polarized beam has a B1
phase modulation caused by the unknown optical plate. The Jones matrix of s polarized beam is
through optical elements in path, the Jones vector reached CCD camera is E1 =
E1 =
7
J8−i E1
i=1
1 0 , After transmission 0 0
A1 , and it is simply expressed as: B1
(1)
Y. Zhao et al. / Optik 127 (2016) 8841–8846
8843
where J7 , J6 · · ·J1 are given by
⎡
⎤
⎢ J7 = ⎣
1 sin 2 2 ⎥
cos2
0
0
⎦ ; J6 = 1 0 1 sin 2 sin2 2 1 −i 1 0 1 ; J1 = J2 = √ 2 −i 1 0 0
ei ; J5
0
0
0
1
=
1 ; J4 = √ 2
1
−i
−i
1
−1
0
0
−1
; J3 =
;
J7 , J6 · · ·J1 are the Jones matrices of polarized optical elements in the passage of s polarized beam, and ϕ is phase and modulation caused by the unknown optical plate. is the angle between transmission axis of P2(polarizer) and X-axis. The Jones matrix is calculated as:
E1 =
⎡
7
J8−i E1 = ⎣
⎤
A1 i sin 2 2 ⎦ ei
(2)
2
A1 isin
i=1
The matrix of p polarized beam is
0 0 . After passing optical elements,the matrix becomesE2 = 0 1
A2 . and it is simply B2
expressed as:
E2 =
7
(3)
J8−i E1
i=1
where J7 , J6 · · ·J1 are given by
⎡ ⎢ J7 = ⎣
⎤
1 sin 2 2 ⎥
cos2
1 0
⎦ ; J6 =
1 0 sin2 sin 2 2 1 −i 0 0 1 J2 = √ ; J1 = 2 −i 1 0 1
1 0
; J5 = 0
0
0
1 ; J4 = √ 2
1
−i
−1
0
; J3 = −i
1
2 i
e 0
(2)
;
−1
J7 , J6 · · ·J1 are the Jones matrices of polarized optical elements in the passage of p polarized beam, where is phase shift caused by PZT shifter with M2. The matrix is calculated as:
⎡
B1 icos2
⎤
2 (2) i ⎦e E2 = ⎣ B i 1 sin 2 2
(4)
After recombining the two beams, the formula is given by:
⎡
⎤ 2 (2) ⎢ A1 sin 2ei + 2B1 cos2 e ⎥ i
E = E1 + E2 =
i 2
⎢ ⎣
2 (2) i 2A1 sin2 ei + B1 sin 2e
⎥ ⎦
(5)
In general, is equal to 45◦ , so the intensity fringe at the CCD camera is calculated as: I = |E | = 2
1 1 2 4 A1 + B12 + A1 B1 cos − 4 2
(6)
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Y. Zhao et al. / Optik 127 (2016) 8841–8846
Fig. 2. Interferograms under different PZT shifter driving voltages. (a)0 V, (b)0.5 V, (c)1.0 V, (d)1.5 V, (e)2.0 V, (f)2.5 V.
If values are 0, /8, /4, 3/8, the intensity fringes are expressed as: I1 =
1 1 2 A1 + B12 + A1 B1 cos 4 2
I2 =
1 1 2 A1 + B12 + A1 B1 sin 4 2
I3 =
1 1 2 A1 + B12 − A1 B1 cos 4 2
I1 =
1 1 2 A1 + B12 − A1 B1 sin 4 2
(7)
ϕ can be quantitatively reconstructed using the formula: ϕ = tan−1
I4 − I2 I3 − I1
(8)
3. Experiment A computer program is employed to control PZT shifter driving controller and CCD camera. The PZT drive voltage step is 0.01 V, and 14 V is the upper boundary. 1400 pieces 600 × 600 pixel interferograms were captured by CCD camera within the range of 0–14 V. Fig. 2 shows interferograms under different PZT shifter driving voltages. The central fringe of each interferogram is marked with a rectangular box. When the voltage is increased to 2.5 V, the marked area has been appeared right horizontal movement. The rectangular box moves from 200× axis pix position to near 400× axis pix position. This indicates that the total interference fringes have displacement caused by the PZT shifter. Throughout the OPSI system, the phase shifting is introduced by the PZT shifter, so the relationship between PZT driving voltage and the phase variety reflects the characteristics of the PZT. Interferograms of each frame are processed by antiGaussian filter and median filter [10–12], and a sine fitting algorithm is used. The initial phases of 1400 pieces interferograms were calculated. The property curve between voltage and phase of PZT shifter is drawn in Fig. 3. In order to verify the testing result of OPSI, an unknown optical plate is tested by OPSI and a common commercial interferometer WYKO [11–14]. The result measured by WYKO is presented as a standard to analyze the error and accuracy of OPSI. The OPSI structure of the experimental setup is shown in Fig. 4. The PZT driving voltages required at Eq. (7) can be got from Fig. 3, and they are 0 V, 2.027 V, 3.380 V, 5.465 V realizing 0, /8, /4, 3/8 phase shift. After capturing 4 pieces interferograms at 0 V, 2.027 V, 3.380 V, 5.465 V,the wrapped phase can be calculated by Eq. (8).The unwrapped phase of the unknown optical plate is obtained by least squares method shown in Fig. 5(b). Fig. 5 shows the results measured by WYKO and OPSI using the unknown optical plate. The phase distribution of the optical plate shown in Fig. 5(a) is similar to Fig. 5(b).This illustrates the accuracy of testing results by OPSI. Errors are common in practice for various reasons including
Y. Zhao et al. / Optik 127 (2016) 8841–8846
8845
Fig. 3. PZT driving voltage and phase shifting curve.
Fig. 4. OPSI experimental setup.
Fig. 5. (a) The phase distribution of optical plate measured by WYKO (b)the phase distribution of optical plate measured by OPSI.
nonlinear and irregular ones [13]. Table 1 is given the detail data measured by OPSI and WYKO, and the error of PV value and RMS value is calculated.
8846
Y. Zhao et al. / Optik 127 (2016) 8841–8846
Table 1 OPSI and WYKO interferometer testing data comparison.
PV value/ RMS value/ PV value error/ RMS value error/
WYKO interferometer
OPSI
0.0342 0.0026 0.0087 0.0006
0.0255 0.0032
4. Conclusion To summarize, an optical interferometry named OPSI(orthogonal phase-shifting interferometry) with common path structure was proposed. And a reconstructed phase method was calculated by Jones matrix. The results of the optical plate experiment show that the OPSI system has high accuracy compared with WYKO, and it can stay good performance at complex environment. It proves a conceptual correctness and feasibility of the proposed method and its applicability to quantitative high-resolution phase measurements. Acknowledgements This work is supported by National Natural Science Foundation of China (NSFC) under Contract No.61308062. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R. Windecker, H.J. Tiziani, Optical roughness measurements using extended white-light interferometry, Opt. Eng. 38 (1999) 1081–1087. S. Han, E. Novak, J. Sullivan, Application of grazing incidence interferometer to rough surface measurements, Proc. SPIE 5144 (2003) 391–400. M.H. Mahdieh, T. Nazari, Measurement of impurity and temperature variations in water by interferometry technique, Optik 125 (2013) 4393–4396. Y.P. Kumar, S. Chatterjee, Thickness measurement of transparent glass plate using a lateral shearing cyclic path optical configuration setup and polarization phase shifting interferometry, Appl. Opt. 49 (2010) 6552–6557. C. Poilâne, P. Sandoz, Thickness measurement of nontransparent free films by double-side white-lightinterferometry: calibration and experiments, Rev. Sci. Instrum. 77 (2006) 055102. R. Bouchal, R. Celechovsky, Z. Bouchal, Polarization sensitive phase-shifting Mirau interferometry using a liquid crystal variable retarder, Opt. Lett. 40 (2015) 4567–4570. L. Zhuo, X.X. Lu, An effective method in calibrating nonlinear errors for phase-shifting interferometry, Optik 125 (2014) 2109–2112. T. Guo, F. Li, J.P. Chen, X. Fu, X.T. Hu, Multi-wavelength phase-shifting interferometry for micro-structures measurement based on color image processing in white light interference, Opt. Lasers Eng. 82 (2016) 41–47. E. Massa, G. Mana, J. Krempel, M. Jentschel, Polarization delivery in heterodyne interferometry, Opt. Express 21 (2013) 27119–27126. J. Militky, M. Kadulova, P. Hlubina, Highly sensitive displacement measurement based on spectral interferometry and Vernier effect, Opt. Commun. 366 (2016) 335–339. Y.P. Kumar, S. Chatterjee, S.S. Negi, Small roll angle measurement using lateral shearing cyclic path polarization interferometry, Appl. Opt. 55 (2016) 979–983. A. Sanchez-Castillo, S. Eslami, F. Giesselmann, P. Fischer, Circular polarization interferometry: circularly polarized modes of cholesteric liquid crystals, Opt. Express 22 (2014) 31227–31236. Z. Sarac, R. Grob, C. Richter, B. Wiesner, G. Hausler, Optimization of white light interferometry on rough surfaces based on error analysis, Optik 115 (2004) 351–357.