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Dual-channel interferometer for vibration-resistant optical measurement
Optics and Lasers in Engineering 127 (2020) 105981
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Dual-channel interferometer for vibration-resistant optical measurement Mingliang Duan a, Yi Zong a, Yixuan Xu a, Guolian Chen a, Wenqian Lu a, Cong Wei a, Rihong Zhu b, Lei Chen a,b, Jianxin Li a,b,∗ a b
School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China MIIT Key Laboratory of Advanced Solid Laser, Nanjing University of Science and Technology, Nanjing 210094, China
a r t i c l e Keywords: Interferometry Dual-channel Vibration resistant
i n f o
a b s t r a c t A dual-channel interferometer (DCI) is proposed for vibration-resistant optical measurement. The DCI employs a dual-channel interference optical path design, wherein an assistant channel is used to generate dense fringe patterns by introducing a spatial carrier frequency, and the coefficients of the vibration tilt phase plane are then extracted. The sparse fringe patterns of the main channel are combined with the coefficients of the tilt phase plane to calculate the measured phase distribution. This study reports the optical path and principles of the interference system. Simulations are performed using the proposed DCI phase extraction algorithm. To verify the DCI performance, phase measurements are carried out in a vibrating environment, and the results obtained using the proposed method are compared with those obtained using spatial phase-shifting interferometry (SPSI). The experimental results demonstrate that the DCI can effectively overcome the effect of vibration. Unlike spatial carrier frequency interferometry, although the spatial carrier frequency is also introduced in the assistant channel in the proposed DCI method, the dual-channel information extraction algorithm helps eliminate the calibration process of the retrace error before measurement, making this a calibration-free measurement. Moreover, the DCI utilizes environmental vibration to generate a ‘phase shift’, thus not only eliminating the need for a vibration isolation platform but also the need for a precise phase shifting device, with advantages such as structural simplicity, ease of operation, and low cost.
1. Introduction Phase-shifting interferometry (PSI) has been extensively used in various interference systems as a noncontact measurement technology for precise optical component surfaces [1]. In conventional PSI, the measured phase distribution is calculated by recording three or more fringe patterns with a phase shift of 𝜋/2 between adjacent frames [2]. PSI requires a phase shift of 𝜋/2 to ensure high measurement accuracy; however, in practical applications, environmental vibrations and air turbulence may cause deviations in the phase shift, leading to fringe errors in the measured phase [3–6]. In recent years, researchers have proposed several interference systems and algorithms to address vibration and air turbulence problems. When the phase shift is assumed to be uniform across the entire interferogram, the phase distribution can be generally obtained using a phaseshifting calibration algorithm or a random phase-shifting algorithm [7– 18]. However, environmental vibrations involve piston as well as tilt vibrations in general measurement conditions. Considering the characteristics of these environmental vibrations, researchers have proposed various phase extraction algorithms that can simultaneously deal with
∗
both piston and tilt vibrations. Chen et al. proposed an iterative algorithm that is unaffected by tilt phase-shift errors [19]. This algorithm replaces the original nonlinear phase error equations with a first-order Taylor series expansion and then determines the phase shift plane by iterative processing. Xu et al. modified the advanced iterative algorithm (AIA) [20] and proposed an iterative algorithm that divides the interferogram into small parts and accurately calculates the phase shifts of the local region. Xu et al. proposed another method to solve tilt vibration utilising spatial and time-domain demodulations [21]. Vargas et al. determined the phase distribution from the tilt phase [22]. Liu et al. proposed a least-squares iterative algorithm to calculate the phase distribution and tilt phase-shift in the x and y directions in three separate least-squares fitting steps [23]. Zeng et al. proposed a regularisation method to extract the phase from fringe patterns with an unknown tilt phase shift [24]. Li et al. proposed phase-tilting interferometry (PTI) for optical testing [25], whereby the tilt vibration can be determined based on line detection. Liu et al. proposed PSI that includes two normalised interferograms with a random tilt phase shift [26]. The algorithm determines the tilt phase shift by extracting the tilt phase-shift plane from the phase difference between two normalised interferograms.
Corresponding author at: School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.optlaseng.2019.105981 Received 12 September 2019; Received in revised form 28 November 2019; Accepted 3 December 2019 0143-8166/© 2019 Elsevier Ltd. All rights reserved.
M. Duan, Y. Zong and Y. Xu et al.
In addition to applying phase-shift calibration and detection algorithms to deal with vibration, some researchers eliminated the effects of vibration by modifying the interference path. Spatial carrier frequency interferometry can be used to perform single-frame phase measurements; this method reduces the effect of vibration to a greater extent. However, it requires introducing a large amount of tilt between the reference and test beams to obtain a carrier-frequency fringe pattern with dense strips. This generates a large non-common optical path retrace error [27]. To reduce the influence of the retrace error on the measurement accuracy, the retrace error must be calibrated for each of the elements to be measured [28]. This calibration process is complicated, affecting the measurement accuracy and efficiency. Smythe and Moore proposed a synchronous phase-shifting interferometry method [29]. In this method, four cameras are used to simultaneously acquire four phase-shifted interferograms, and a four-step phase shift method is used to calculate the measured phase distribution. This technique requires calibrating the position and angle of each camera, resulting in a more complex device setup. Millerd et al. developed a device for simultaneously acquiring four phase-shifted interferograms on a camera [30,31]. Although this technique solves the problems of camera alignment and synchronisation acquisition, it requires high-precision spectroscopic gratings and polarisation elements. Novak et al. proposed a pixelated phase-mask dynamic interferometer [32], which has a good performance in terms of vibration resistance and testing precision. However, the polarised detector technique is relatively complicated. Another technique for vibration-resistant interference involves the detection and compensation of the vibration signal. Zhao and Burge proposed a vibration-compensated interferometer for surface metrology [33]. This technique uses a high-speed point sensor to detect real-time vibrations, which are then used for calibrating the phase difference between the reference and test beams; however, this method is limited to solving piston vibration. Broistedt et al. proposed a modified method [34] using three high-speed point sensors to detect piston as well as tilt vibration in real time. However, both methods require a high-speed loop feedback and are based on point detection. Recently, we proposed a phase-deforming interferometry (PDI) method that effectively overcomes the effects of vibration and air turbulence [35]. However, its optical path is more complicated, and because of the structural characteristics of its carrier-frequency pass, the reference and test beams of the carrier frequency pass have a significant lateral deviation on the detector target surface when a high carrier frequency is introduced. Therefore, it is necessary to reduce the test beam diameter or increase the diameter of the reference beam or replace the detector of the larger target surface; this results in inefficient use of the spatial resolution of the detector. Furthermore, in the PDI method, the tilt phase is calculated using Fourier transform interferometry; the Gibbs effect observed in Fourier transform interferometry will reduce the accuracy of the tilt phase. In this study, we propose a dual-channel interferometer (DCI) for vibration-resistant optical measurement. The proposed DCI can be used to accurately detect the tilt phase plane using dual-channel interferometry, which is a novel concept in interferometry, and the measured phase distribution can then be determined. Conventional spatial carrier frequency interferometry requires introducing a large tilt between the reference and test beams, thus generating a large non-common optical path retrace error. The calibration process of the retrace error is complicated, and it affects the measurement accuracy and efficiency. Similar to spatial carrier frequency interferometry, the spatial carrier frequency is also introduced in the assistant channel in the proposed method; therefore, the interferograms of this channel cannot be used to accurately extract the measured phase. However, in the DCI method, the assistant channel is only used to obtain the difference in the coefficients of the tilt phase plane relative to the first interferogram, thus avoiding the influence of the retrace error. The new concept of dual-channel information extraction in the DCI helps eliminate the calibration process of the retrace error before measurement, making this a calibration-free measurement.
Optics and Lasers in Engineering 127 (2020) 105981
Furthermore, we propose a new algorithm, namely the stepwise local up-sampling phase correlation algorithm, to extract the coefficients of the tilt phase plane directly from the interferogram spectrum of the assistant channel. This eliminates the process of solving the tilt phase using the Fourier method and phase unwrapping. The remainder of this paper is organised as follows. In Section 2, the principle of the DCI is described in detail. The DCI phase extraction technique is introduced in Section 3. The results of simulations performed using the proposed dual-channel phase extraction algorithm are reported in Section 4. The results of experiments conducted to verify the vibration resistance performance of the DCI, including a comparison with spatial phase-shifting interferometry (SPSI), are presented in Section 5. The obtained results are discussed in Sections 6 and 7 concludes the study. 2. Optical principle of a dual-channel interferometer Fig. 1 shows the schematic of the proposed DCI. The interference system is divided into three parts: a light source and main- and assistantchannel interference optical paths. The light source comprises a laser, collimating lenses L1 and L2, and a half-wave plate (HWP). Fig. 2(a) shows the schematic of the main channel. After passing through a polarisation beam splitter (PBS), the laser beam can be divided into transmitted and reflected beams, i.e. p and s waves, respectively. After passing through a 𝜆/4 wave plate (QWP1), the s wave is reflected by a reference flat (RF1), after which it passes through QWP1 again, the fastaxis of which is at an angle of 45° with the s wave. The polarisation direction of the s wave is rotated by 90° after passing through QWP1 twice, and this wave becomes the reference beam of the main channel. The p wave transmitted from the PBS passes through another 𝜆/4 wave plate (QWP2) and divergence (L3), subsequently passing through QWP2 again after being reflected by a test sphere (TS). The fast-axis of QWP2 is at an angle of 22.5° with the p wave. After passing through QWP2 twice, the polarisation direction of the p wave is rotated by 90°; this wave becomes the test beam. The beam is then split by the PBS into a transmitted p wave and a reflected s wave, and the reflected beam becomes the test beam of the main channel. The test and reference beams of the main channel are orthogonal and linearly polarised and are synthesised using a polaroid (P3) to form an interferogram on camera (D) after passing through a mirror (RM) and objective lenses L4 and L5. Fig. 2(b) shows the schematic of the assistant channel. The reflected beam from the beam splitter (BS1) passes through a polaroid (P1) and a standard reference flat (RF2), becoming the reference beam of the assistant channel. The p wave returning from the PBS of the main channel is the test beam of the assistant channel. After being synthesised by the polaroid (P2), the reference and test beams pass through the objective lenses L4 and L5 to form an interferogram on D. In the DCI, the reference flats, RF1 and RF2, are mounted in one frame to ensure synchronous vibration of the two channels. A large carrier frequency is introduced in the assistant channel by separately adjusting the tilt angles of RF1 and RF2, while that of the main channel remains small. During testing, camera D simultaneously acquires a series of interferograms of the two channels. The interferogram sequence of the assistant channel is used for solving the coefficients of the vibration tilt phase plane, whereas the interferogram sequence of the main channel is used for solving the measured phase distribution by combining the tilt phase plane obtained from the assistant channel. The process of solving the two-channel phase is given in Section 3. 3. Phase extraction algorithm In the presence of vibration, the interference intensity of the main channel is as follows: 𝐼𝑛 (𝑥, 𝑦) = 𝑎(𝑥, 𝑦) + 𝑏(𝑥, 𝑦) cos(𝜑(𝑥, 𝑦) + 𝑃𝑛 (𝑥, 𝑦)),
(1)
M. Duan, Y. Zong and Y. Xu et al.
Optics and Lasers in Engineering 127 (2020) 105981
Fig. 1. Schematic of a dual-channel interferometer (DCI).
Fig. 2. Flat schematics of the DCI: Optical path of the (a) main channel and (b) assistant channel.
where𝑛 = 1, 2, ⋅ ⋅ ⋅, 𝑁, a(x, y)is the background intensity, b(x, y) is the modulation amplitude, 𝜑(x, y) is the phase map, and Pn (x, y) is the vibration tilt phase plane. Pn (x, y) is defined as follows: 𝑃𝑛 (𝑥, 𝑦) = 𝛼𝑛 𝑥 + 𝛽𝑛 𝑦 + 𝛾𝑛 .
(2)
According to the structural characteristics of the optical system described in Section 2, RF1 and RF2 are integral, and there is a tilt angle between them. Therefore, the tilt phase planes of the main and assistant channels are the same. Thus, the interference intensity expression for the assistant channel is as follows: 𝐼𝑛′ (𝑥, 𝑦) = 𝑎′ (𝑥, 𝑦) + 𝑏′ (𝑥, 𝑦) cos(𝜑(𝑥, 𝑦) + 𝐸(𝑥, 𝑦) + 𝑃𝑛 (𝑥, 𝑦)),
obtaining Pn (x, y). The measured phase 𝜑(x, y) can then be obtained from the main-channel fringe patterns using the main channel phase extraction algorithm described in Section 3.2.
(3)
Here, E(x, y) is the carrier-frequency phase and retrace error introduced by the tilt of RF2. In Eqs. (1) and (3), we assume that 𝑃𝑛 (𝑥, 𝑦) = 0. To solve the phase map 𝜑(x, y), the assistant-channel fringe patterns are first processed using the tilt phase plane coefficient extraction algorithm, described in Section 3.1, to determine 𝛼 n , 𝛽 n , and 𝛾 n , thereby
3.1. Extraction algorithm for tilt phase plane coefficients To obtain the tilt phase plane, we propose a tilt phase plane coefficient extraction algorithm. In this method, a Fourier transform is performed on the dense fringe pattern of the assistant channel, and the carrier-frequency coefficients (defined as kxn , kyn , andkn ) of each fringe pattern are directly calculated in the frequency domain. The tilt phase plane coefficients 𝛼 n ,𝛽 n , and 𝛾 n can be obtained from kxn , kyn , andkn . The proposed algorithm can extract the tilt phase plane coefficients quickly and accurately without having to solve the phase of the interferogram, thus eliminating the process of solving the phase using the
M. Duan, Y. Zong and Y. Xu et al.
Optics and Lasers in Engineering 127 (2020) 105981
Fourier method and phase unwrapping and avoiding the Gibbs effect. This significantly improves the efficiency and accuracy of measurement. Eq. (3) can be rewritten in a complex form as follows: 𝐼′
𝑛 (𝑥, 𝑦)
𝑎′ (𝑥, 𝑦) +
= exp(𝑗(𝑘𝑥𝑛 𝑥 + 𝑘𝑦𝑛 𝑦 + 𝑘𝑛 ))𝑐(𝑥, 𝑦) + exp(−𝑗(𝑘𝑥𝑛 𝑥 + 𝑘𝑦𝑛 𝑦 + 𝑘𝑛 ))𝑐 ∗ (𝑥, 𝑦),
(4)
where kxn , kyn , and kn are the carrier-frequency coefficients of the assistant channel, 𝑐(𝑥, 𝑦) = 1∕2𝑏′ (𝑥, 𝑦) exp(𝑗 𝜑𝑛 (𝑥, 𝑦)), and c∗ (x, y) is the conjugate of c(x, y). We define: Γ(𝑐(𝑥, 𝑦)) = 𝐶(𝑥, 𝑦),
(5)
3.2. Main channel phase extraction algorithm The retrace error can be neglected because of the small number of fringes in the main-channel interferograms. Thus, the measured phase 𝜑(x, y) can be calculated by combining the tilt phase plane obtained from the assistant channel with the main-channel interferograms. Defining, 𝑐1 (𝑥, 𝑦) = 𝑎(𝑥, 𝑦) 𝑐2 (𝑥, 𝑦) = 𝑏(𝑥, 𝑦) cos(𝜑(𝑥, 𝑦)) , 𝑐3 (𝑥, 𝑦) = −𝑏(𝑥, 𝑦) sin(𝜑(𝑥, 𝑦))
(11)
where Γ( · ) is the Fourier transform operation. By performing a 2D Fourier transform on the interferogram of size MA × NA of the assistant channel, we can obtain the following.
and omitting (x, y) for convenience, Eq. (1) can be rewritten as follows:
𝐹 (𝑢, 𝑣) = Γ(𝐼 ′ 𝑛 (𝑥, 𝑦)) = 𝐴′ (𝑢, 𝑣) + exp(𝑗 𝑘𝑛 )𝐶(𝑢 + 𝑓𝑥 , 𝑣 + 𝑓𝑦 ) + exp(−𝑗 𝑘𝑛 )𝐶 ∗ (𝑢 − 𝑓𝑥 , 𝑣 − 𝑓𝑦 ),
We denote the measured interferograms as 𝐼𝑛′′ (𝑥, 𝑦). The sum of the squared differences between the measured and theoretical interferograms predicted using Eq. (12) is given by ∑𝑁 2 𝑆= (𝐼𝑛 − 𝐼 ′′ 𝑛 ) , (13)
(6)
where A′(u, v) is the spectrum of the background intensity, and fx and fy are the coordinates of the positive and negative first-order peaks in the spectrum, respectively: 𝑓𝑥 = 𝑘𝑥𝑛 𝑁𝐴 ∕2𝜋, 𝑓𝑦 = 𝑘𝑦𝑛 𝑀𝐴 ∕2𝜋.
(7)
To extract fx and fy quickly and accurately, we propose the stepwise local up-sampling phase correlation algorithm, which includes a local up-sampling phase correlation algorithm and a stepwise algorithm. This algorithm only applies up-sampling to the adjacent areas of the peak, thus reducing the computation time and saving memory. The local up-sampling phase correlation algorithm implements Fourier transform on partial regions through matrix Fourier transform. The matrix Fourier transform of a 2D discrete signal 𝐼𝑛′ (𝑋, 𝑌 ) can be expressed as follows: ( ) ( ) 𝐹 (𝑈 , 𝑉 ) = exp −2𝜋jU𝑋 𝑇 𝐼𝑛′ (𝑋, 𝑌 ) exp 2𝜋jY𝑉 𝑇 ,
(8)
where 𝑋 = (𝑥0 , 𝑥1 , ⋅ ⋅ ⋅, 𝑥𝑁𝐴 −1 )𝑇 , 𝑌 = (𝑦0 , 𝑦1 , ⋅ ⋅ ⋅, 𝑦𝑀𝐴 −1 )𝑇 , 𝑈= (𝑢0 , 𝑢1 , ⋅ ⋅ ⋅, 𝑢𝑁𝐵 −1 )𝑇 , and 𝑉 = (𝑣0 , 𝑣1 , ⋅ ⋅ ⋅, 𝑣𝑁𝐵 −1 )𝑇 . We first extract the approximate position (fx0 , fy0 ) of the peak using the conventional method and then give the up-sampling area size m(1 ≤ m ≤ 2) and sampling magnification r. The sampling step is 1/r, and the number of sampling points is 𝑁𝐵 = 𝑚𝑟. i indicates the ordinal of the sampling points; accordingly, 𝑢𝑖 = 𝑓𝑥0 − 𝑁𝐵 ∕2 + 𝑖 and 𝑣𝑖 = 𝑓𝑦0 − 𝑁𝐵 ∕2 + 𝑖. The up sampling to the adjacent areas of the peak can be achieved using Eq. (8). When the required positioning accuracy is high, it is necessary to increase the sampling magnification r. To improve the computation speed and save computation memory, it is necessary to use the stepwise algorithm. For example, when 𝑟 = 10000, step 1: take 𝑟1 = 10 and perform the first up sampling according to the above method to locate the peak position of pixel/10 precision; step 2: perform the up sampling again with magnification r1 on the data sampled in step 1 to locate the peak position of pixel/100 precision; step 3: Repeat the above steps until 10,000 times sampling and locate the high-precision peak positions fx and fy . kxn and kyn can then be obtained using Eq. (7). After determining fx and fy , kn can be obtained using Eqs. (6) and (8): ( ( )) ( ( )) 𝑘𝑛 = 𝐴𝑟𝑔 𝐹 𝑓𝑥 , 𝑓𝑦 = 𝐴𝑟𝑔 exp −𝑗𝑘𝑛 ( ( ) ( )) = 𝐴𝑟𝑔 exp −2𝜋𝑗 𝑓𝑥 𝑋 𝑇 𝐼 ′ 𝑛 (𝑋, 𝑌 ) exp 2𝜋𝑗𝑌 𝑓𝑦 ,
(9)
where Arg( · ) is the function used to calculate the phase angle. After obtaining the carrier-frequency coefficients of the assistantchannel interferograms using the above algorithm according to the equation 𝛼𝑛 = 𝑘𝑥𝑛 − 𝑘𝑥1 , 𝛽𝑛 = 𝑘𝑦𝑛 − 𝑘𝑦1 , 𝛾𝑛 = 𝑘𝑛 − 𝑘1 ,
𝐼𝑛 = 𝑐1 + 𝑐2 cos(𝑃𝑛 ) + 𝑐3 sin(𝑃𝑛 ).
(12)
𝑛=1
where N is the total number of interferograms, and c1 , c2 , and c3 are three unknowns. The unknown variables can be determined by minimising S using the least-squares criterion: 𝜕𝑆 ∕𝜕 𝑐1 = 0, 𝜕𝑆 ∕𝜕 𝑐2 = 0, 𝜕𝑆 ∕𝜕 𝑐3 = 0.
(14)
The solutions to the above three equations are given by the following linear matrix equations: 𝐶 = 𝐴−1 𝐵, where [ 𝐶 = 𝑐1 𝐵=
(15)
𝑐2
[ ∑ 𝑁
𝑛=1
]T
𝑐3
𝐼 ′′ 𝑛
∑𝑁
𝑁 ⎡ ∑ 𝐴=⎢ 𝑁 cos(𝑃𝑛 ) ⎢ ∑𝑛𝑁=1 ⎣ 𝑛=1 sin(𝑃𝑛 )
,
𝑛=1
(16) 𝐼 ′′ 𝑛 cos(𝑃𝑛 )
∑𝑁
𝑛=1
∑𝑁 cos(𝑃 ) ∑𝑁𝑛=1 2 𝑛 cos (𝑃𝑛 ) ∑𝑁 𝑛=1 ′ 𝑛=1 sin(𝛿 𝑛 ) cos(𝑃𝑛 )
𝐼 ′′ 𝑛 sin(𝑃𝑛 )
]𝑇
,
(17)
∑𝑁 sin(𝑃 ) ⎤ ∑𝑁 𝑛=1 ′ 𝑛 ⎥ cos ( 𝛿 𝑛 ) sin(𝑃𝑛 )⎥. (18) 𝑛=1 ∑𝑁 2 ⎦ 𝑛=1 sin (𝑃𝑛 )
After determining c1 and c2 , we can obtain the phase map, i.e. 𝜑(x, y), as follows: 𝜑 = tan−1 (−𝑐3 ∕𝑐2 ).
(19)
4. Simulations To verify the feasibility and accuracy of the DCI phase extraction algorithm, we carried out simulation experiments. In the experiment, the test phase is fitted using the Zernike function. Table 1 lists the fitting coefficients. According to the tilt plane coefficients, listed in Table 2, eight groups of the main-channel and assistant-channel interferograms are simulated, and the resolution of the interferogram is 512 × 512. Fig. 3(a)–3(g) show the main-channel interferograms. Fig. 3(h) shows the assistant-channel interferogram with a high carrier frequency. The coefficients of the tilt phase planes can be obtained using the proposed tilt phase plane coefficient extraction algorithm. We set the sampling magnification 𝑟 = 10000; therefore, the location precision of the peak is pixel/10,000. Table 2 lists the coefficients. The maximum errors of the calculated tilt plane coefficients 𝛼 n , 𝛽 n , and 𝛾 n with respect to the real value are 5 × 10−6 rad/pixel, 3 × 10−6 rad/pixel, and 0.0255 rad. Table 1 Zernike Coefficients of the test phase for the simulation.
(10)
we can obtain the tilt phase plane generated using the coefficients 𝛼 n , 𝛽 n , and𝛾 n .
Zernike Terms
1
2
3
4
Coefficients
0.03
0.02
0.15
0.30
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Optics and Lasers in Engineering 127 (2020) 105981
Table 2 Extracted results of tilt phase plane. Frame
2 3 4 5 6 7 8
𝛼 (×10−2 rad/pixel)
𝛽 (×10−2 rad/pixel)
𝛾 (rad)
Real
Calculated
Real
Calculated
Real
Calculated
−2.0783 −0.4423 −1.5128 −1.2635 −1.8058 −0.5167 −1.1404
−2.0786 −0.4423 −1.5129 −1.2639 −1.8059 −0.5175 −1.1409
−0.6727 1.2799 0.0754 0.6520 2.0253 −0.2166 −0.2048
−0.6727 1.2800 0.0753 0.6518 2.0256 −0.2167 −0.2046
1.6514 4.6139 3.2902 2.5300 5.9449 1.4650 4.4651
1.6478 4.6091 3.3177 2.5555 5.9456 1.4634 4.4452
This proves the high precision of the proposed local up-sampling phase correlation algorithm. Fig. 4(a) and 4(b) show the real test phase and calculated phase extracted using the proposed method: the peak-tovalley (PV) values are 0.5977𝜆 and 0.5980𝜆, respectively, and the root mean square (RMS) values are 0.126490𝜆 and 0.126497𝜆, respectively. Fig. 4(c) shows the residual error between the results shown in Figs. 4(a) and 4(b): PV = 0.003𝜆 and RMS = 5.5325 × 10−4 𝜆. The simulation results prove that the proposed DCI phase extraction algorithm is feasible and accurate. 5. Experiments and analysis 5.1. Experimental setup
Fig. 3. Interferograms obtained from simulation.
To verify the feasibility of the DCI, we constructed an experimental setup based on the optical path principle. The light-source includes an HeNe laser with a wavelength of 𝜆 = 632.8 nm. The divergence, L3, has an F number of 5 and a focal length of 50 mm. The optical element to
Fig. 4. Simulation results: (a) real phase, (b) calculated phase, (c) residual error.
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Optics and Lasers in Engineering 127 (2020) 105981
Fig. 5. Results of the comparative experimental: (a) four phase-shifting interferograms, (b) phase obtained by four-step phase-shifting interferometry.
be measured is a spherical mirror having a diameter of 25 mm and a curvature radius of 125 mm. The focal lengths of the imaging lenses L2 and L3 are 150 and 75 mm, respectively. The detector is a CMOS camera (Point Grey, GS3-U3-23S6M-C) with a resolution of 1920 × 1200 and a pixel size of 5.8 𝜇m. Two standard mirrors are attached onto the same substrate to ensure synchronous vibration. In the following experiment, the ambient temperature is 23 ◦ C, and the vibration frequency is in the range of 20–200 Hz. The experimental environment is generally representative.
5.2. Experiments in vibrating environment To verify the performance of the DCI with respect to environmental vibration resistance, we conducted experiments in a vibrating environment. In the vibrating environment, 20-frame interferograms were acquired. The number and direction of the fringes varied considerably. Firstly, Using the four-step phase-shifting interferometry, we calculated the measured phase from the first four interferograms of the main channel. Figs. 5(a) and (b) show the four phaseshifting interferograms and calculation results. It can be seen from Fig. 5(b) that there are obvious ripple errors caused by vibration, with PV = 0.8109𝜆 and RMS = 0.0827𝜆. And then using the previously mentioned algorithm, we obtained the measured phase from the acquired interferograms. Figs. 6(a) and (b) show the three-frame interferograms of the main and assistant channels. The measured phase was extracted using the DCI phase extraction algorithm. Fig. 6(c) and 6(d) show the results: PV = 0.3471𝜆 and RMS = 0.0578𝜆. The experimental results show that the phase extracted by DCI method has no ripple error, despite performing the experiment in a vibrating environment, thus proving the excellent vibration-resistance performance of the DCI. To verify the reliability of the DCI measurements, we conducted comparative experiments with SPSI. To simultaneously perform DCI and spatial phase-shifting measurements in the same optical path, we replaced the polarising plate P3 in the main channel with a 𝜆/4 wave plate QWP3 and the camera with a polarisation camera (FLIR, BFS-U351S5P-C, number of pixels: 2448 × 2048). In this experimental setup, only two devices were replaced. Thus, the system errors and aberration
in the two schemes were the same during the experiment, and the experimental results were more comparable. The polarisation camera captures 20-frame fringe patterns, each of which can be divided into four interferograms with a phase shift of 𝜋/2. For the DCI method, we divided all the fringe patterns into four interferograms and then calculated the measured phase using the interferograms with the same phase shift. Fig. 8(a) shows the phase results: PV = 0.3321𝜆 and RMS = 0.0599𝜆; For the SPSI method, we divided one fringe pattern into four interferograms and extracted the region of the main channel. Fig. 7 shows the interferograms. Fig. 8(b) shows the phase distribution calculated using the four-step phase-shifting method: PV = 0.3373𝜆 and RMS = 0.0595𝜆. Fig. 8(c) shows the residual error between the phase extracted using the DCI and SPSI: PV = 0.1002𝜆 and RMS = 0.0093𝜆. The phase results of the two methods are consistent. A small ripple error equal to the frequency of the fringe patterns can still be seen in the residual error phase, because of the inconsistent background intensity of the four spatial phase-shifting interferograms. Thus, the experimental results demonstrate the reliability of the DCI measurements. 6. Discussion The experiments conducted in the study verify the effectiveness and reliability of the DCI with respect to vibration resistance. Considering the optical path, system devices, and experimental results, the DCI can be beneficial and practical for optical testing in vibration environments. (i) Similar to the spatial carrier frequency interferometry, a high carrier frequency is also introduced in the assistant channel in the DCI method, thus generating a large non-common optical path retrace error and a systematic error in this channel. Therefore, the interferograms of this channel cannot be used to extract the final phase. However, the assistant channel is only used to obtain the difference in the coefficients of the tilt phase plane relative to the first interferogram. Moreover, as the vibration of the two channels is synchronous, the tilt phase plane generated based on the coefficients of the assistant channel can reflect the vibration of the main channel. Finally, the measured phase is extracted using the interferograms of the main channel and the tilt phase plane, with no influence of the retrace
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Fig. 6. Results of the vibration-resistant experiments: (a) interferograms of the main channel, (b) interferograms of the assistant channel, (c) 2D and 3D phase maps.
Fig. 7. Spatial phase-shifting interferograms extracted by a polarization camera.
error. In order to reduce the non-common optical path retrace error of the main channel and obtain more accurate coefficients of the tilt phase plane from the assistant channel, the number of fringe of the two channels should be within 5 and between 100 and 200 respectively. The new concept of dual-channel information extraction in the DCI helped eliminates the calibration process of retrace error before measurement, thus simplifying the measurement process and making it easier to apply the system to actual measurements.
(ii) Compared with the PDI method proposed by our research group recently, the DCI method has some advantages. The role of the carrier frequency pass in the PDI is taken up by the assistant channel in the DCI; this simplifies the optical system and operation process. The reference and test beams of the assistant channel have a smaller lateral deviation on the detector target surface when a high carrier frequency is introduced. Thus, the spatial resolution of the detector can be better utilized. Furthermore, the proposed stepwise local up-sampling phase correlation algorithm can extract the coefficients of the tilt phase plane directly from the interferogram spectrum of the assistant channel. This eliminates the process of solving the tilt phase using the Fourier method and phase unwrapping and avoids the Gibbs effect, thus significantly improving the efficiency and accuracy of measurement. There are also some disadvantages for the DCI compared with the PDI. In the DCI, the reference beams of the main channel and assistant channel are generated by the two reference flats respectively. Although the two reference flats are mounted in the same structure, the structural stability will still decrease after
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Fig. 8. Results of the comparative experiments: (a) phase map extracted using the DCI, (b) phase map extracted using the SPSI, (c) residual error.
long-term use, which will affect the vibration synchronicity of the two channels and cause errors in measurement results. (iii) The DCI efficiently uses two interference channels to directly determine the vibration tilt phase plane; this is a novel technique in interferometry. Although the two channels are both off-axis imaged, the beams of them are collimated and normal incidence on the camera and symmetrical about the optical axis. So off-axis imaging will not affect the test results. Environmental vibration was utilized to perform ‘phase shifting’ for the surface measurements of optical components. The DCI measurement method not only eliminates the need for a vibration isolation platform, but also the need for expensive and precise phase-shifting devices. Moreover, as the DCI is of the temporal phase-shifting type, the inconsistency in the contrast between the fringe patterns due to polarisation elements is avoided. (iv) The DCI method has some limitations. Similar to most beam splitting interferometry methods, the proposed DCI method requires calibration between the interferograms of the main and assistant channels. Therefore, the calibration process affects the measurement results.
of the DCI, including a comparison with SPSI. The experimental results demonstrated that the DCI has good vibration resistance performance. Compared with the standard vibration-resistant interferometer, the DCI uses environmental vibration for ‘phase shifting’, does not require a vibration isolation platform or precise phase shifting devices, and has advantages such as structural simplicity, ease of operation, and low cost. Funding National Natural Science Foundation of China (NSFC) (61,975,079), (61,475,072); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0399). Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
7. Conclusions
Supplementary materials
In summary, we proposed a DCI for vibration-resistant optical testing. The DCI employs a dual-channel interference optical path design. An assistant channel is used for calculating the coefficients of the tilt phase plane using the stepwise local up-sampling phase correlation algorithm; the main channel fringe patterns are combined with the tilt phase plane generated based on the coefficients of the assistant channel to obtain the measured phase using the least-squares method. Simulations were performed using the DCI phase extraction algorithm, and experiments were conducted to verify the vibration-resistant performance
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.optlaseng.2019.105981. CRediT authorship contribution statement Mingliang Duan: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Writing - review & editing, Formal analysis. Yi Zong: Software, Formal analysis, Investigation. Yixuan Xu: Resources, Data curation. Guolian Chen: Software,
M. Duan, Y. Zong and Y. Xu et al.
Data curation. Wenqian Lu: Software, Data curation. Cong Wei: Software, Resources. Rihong Zhu: Supervision, Project administration. Lei Chen: Resources, Project administration. Jianxin Li: Conceptualization, Methodology, Resources, Supervision, Project administration. References [1] Malacara D, Servín M, Malacara Z. Interferogram analysis for optical testing. 2nd ed. Taylor and Francis; 2005. [2] Malacara D. Optical shop testing. 3rd ed. CRC Press; 2007. [3] de Groot PJ. Vibration in phase-shifting interferometry. J Opt Soc Am A 1995;12(2):354–65. [4] Deck LL. Suppressing phase errors from vibration in phase-shifting interferometry. Appl Opt 2009;48(20):3948–60. [5] Deck LL. Model-based phase shifting interferometry. Appl Opt 2014;53(21):4628–36. [6] de Groot PJ, Deck LL. Numerical simulations of vibration in phase-shifting interferometry. Appl Opt 1996;35(13):2172–8. [7] Okada K, Sato A, Tsujiuchi J. Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry. Opt Commun 1991;84(3–4):118–24. [8] Wang Z, Han B. Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations. Opt Laser Eng 2007;45(2):274–80. [9] Cai LZ, Liu Q, Yang XL. Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps. Opt Lett 2003;28(19):1808–10. [10] Xu XF, Cai LZ, Meng XF, Dong GY, Shen XX. Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry. Opt Lett 2006;31(13):1966–8. [11] Xu J, Jin W, Chai L, Xu Q. Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method. Opt Express 2011;19(21):20483–92. [12] Vargas J, Sorzano COS, Estrada JC, Carazo JM. Generalization of the principal component analysis algorithm for interferometry. Opt Commun 2013;286:130–4. [13] Vargas J, Sorzano COS. Quadrature component analysis for interferometry. Opt Laser Eng 2013;51:637–41. [14] Liu F, Wu Y, Wu F. Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology. Opt Express 2015;23(8):10794–807. [15] Tay CJ, Quan C, Chen L. Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift. Appl Opt 2005;44(8):1401–9.
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Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Dual-channel interferometer for vibration-resistant optical measurement Mingliang Duan a, Yi Zong a, Yixuan Xu a, Guolian Chen a, Wenqian Lu a, Cong Wei a, Rihong Zhu b, Lei Chen a,b, Jianxin Li a,b,∗ a b
School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China MIIT Key Laboratory of Advanced Solid Laser, Nanjing University of Science and Technology, Nanjing 210094, China
a r t i c l e Keywords: Interferometry Dual-channel Vibration resistant
i n f o
a b s t r a c t A dual-channel interferometer (DCI) is proposed for vibration-resistant optical measurement. The DCI employs a dual-channel interference optical path design, wherein an assistant channel is used to generate dense fringe patterns by introducing a spatial carrier frequency, and the coefficients of the vibration tilt phase plane are then extracted. The sparse fringe patterns of the main channel are combined with the coefficients of the tilt phase plane to calculate the measured phase distribution. This study reports the optical path and principles of the interference system. Simulations are performed using the proposed DCI phase extraction algorithm. To verify the DCI performance, phase measurements are carried out in a vibrating environment, and the results obtained using the proposed method are compared with those obtained using spatial phase-shifting interferometry (SPSI). The experimental results demonstrate that the DCI can effectively overcome the effect of vibration. Unlike spatial carrier frequency interferometry, although the spatial carrier frequency is also introduced in the assistant channel in the proposed DCI method, the dual-channel information extraction algorithm helps eliminate the calibration process of the retrace error before measurement, making this a calibration-free measurement. Moreover, the DCI utilizes environmental vibration to generate a ‘phase shift’, thus not only eliminating the need for a vibration isolation platform but also the need for a precise phase shifting device, with advantages such as structural simplicity, ease of operation, and low cost.
1. Introduction Phase-shifting interferometry (PSI) has been extensively used in various interference systems as a noncontact measurement technology for precise optical component surfaces [1]. In conventional PSI, the measured phase distribution is calculated by recording three or more fringe patterns with a phase shift of 𝜋/2 between adjacent frames [2]. PSI requires a phase shift of 𝜋/2 to ensure high measurement accuracy; however, in practical applications, environmental vibrations and air turbulence may cause deviations in the phase shift, leading to fringe errors in the measured phase [3–6]. In recent years, researchers have proposed several interference systems and algorithms to address vibration and air turbulence problems. When the phase shift is assumed to be uniform across the entire interferogram, the phase distribution can be generally obtained using a phaseshifting calibration algorithm or a random phase-shifting algorithm [7– 18]. However, environmental vibrations involve piston as well as tilt vibrations in general measurement conditions. Considering the characteristics of these environmental vibrations, researchers have proposed various phase extraction algorithms that can simultaneously deal with
∗
both piston and tilt vibrations. Chen et al. proposed an iterative algorithm that is unaffected by tilt phase-shift errors [19]. This algorithm replaces the original nonlinear phase error equations with a first-order Taylor series expansion and then determines the phase shift plane by iterative processing. Xu et al. modified the advanced iterative algorithm (AIA) [20] and proposed an iterative algorithm that divides the interferogram into small parts and accurately calculates the phase shifts of the local region. Xu et al. proposed another method to solve tilt vibration utilising spatial and time-domain demodulations [21]. Vargas et al. determined the phase distribution from the tilt phase [22]. Liu et al. proposed a least-squares iterative algorithm to calculate the phase distribution and tilt phase-shift in the x and y directions in three separate least-squares fitting steps [23]. Zeng et al. proposed a regularisation method to extract the phase from fringe patterns with an unknown tilt phase shift [24]. Li et al. proposed phase-tilting interferometry (PTI) for optical testing [25], whereby the tilt vibration can be determined based on line detection. Liu et al. proposed PSI that includes two normalised interferograms with a random tilt phase shift [26]. The algorithm determines the tilt phase shift by extracting the tilt phase-shift plane from the phase difference between two normalised interferograms.
Corresponding author at: School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China. E-mail address: [email protected] (J. Li).
https://doi.org/10.1016/j.optlaseng.2019.105981 Received 12 September 2019; Received in revised form 28 November 2019; Accepted 3 December 2019 0143-8166/© 2019 Elsevier Ltd. All rights reserved.
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In addition to applying phase-shift calibration and detection algorithms to deal with vibration, some researchers eliminated the effects of vibration by modifying the interference path. Spatial carrier frequency interferometry can be used to perform single-frame phase measurements; this method reduces the effect of vibration to a greater extent. However, it requires introducing a large amount of tilt between the reference and test beams to obtain a carrier-frequency fringe pattern with dense strips. This generates a large non-common optical path retrace error [27]. To reduce the influence of the retrace error on the measurement accuracy, the retrace error must be calibrated for each of the elements to be measured [28]. This calibration process is complicated, affecting the measurement accuracy and efficiency. Smythe and Moore proposed a synchronous phase-shifting interferometry method [29]. In this method, four cameras are used to simultaneously acquire four phase-shifted interferograms, and a four-step phase shift method is used to calculate the measured phase distribution. This technique requires calibrating the position and angle of each camera, resulting in a more complex device setup. Millerd et al. developed a device for simultaneously acquiring four phase-shifted interferograms on a camera [30,31]. Although this technique solves the problems of camera alignment and synchronisation acquisition, it requires high-precision spectroscopic gratings and polarisation elements. Novak et al. proposed a pixelated phase-mask dynamic interferometer [32], which has a good performance in terms of vibration resistance and testing precision. However, the polarised detector technique is relatively complicated. Another technique for vibration-resistant interference involves the detection and compensation of the vibration signal. Zhao and Burge proposed a vibration-compensated interferometer for surface metrology [33]. This technique uses a high-speed point sensor to detect real-time vibrations, which are then used for calibrating the phase difference between the reference and test beams; however, this method is limited to solving piston vibration. Broistedt et al. proposed a modified method [34] using three high-speed point sensors to detect piston as well as tilt vibration in real time. However, both methods require a high-speed loop feedback and are based on point detection. Recently, we proposed a phase-deforming interferometry (PDI) method that effectively overcomes the effects of vibration and air turbulence [35]. However, its optical path is more complicated, and because of the structural characteristics of its carrier-frequency pass, the reference and test beams of the carrier frequency pass have a significant lateral deviation on the detector target surface when a high carrier frequency is introduced. Therefore, it is necessary to reduce the test beam diameter or increase the diameter of the reference beam or replace the detector of the larger target surface; this results in inefficient use of the spatial resolution of the detector. Furthermore, in the PDI method, the tilt phase is calculated using Fourier transform interferometry; the Gibbs effect observed in Fourier transform interferometry will reduce the accuracy of the tilt phase. In this study, we propose a dual-channel interferometer (DCI) for vibration-resistant optical measurement. The proposed DCI can be used to accurately detect the tilt phase plane using dual-channel interferometry, which is a novel concept in interferometry, and the measured phase distribution can then be determined. Conventional spatial carrier frequency interferometry requires introducing a large tilt between the reference and test beams, thus generating a large non-common optical path retrace error. The calibration process of the retrace error is complicated, and it affects the measurement accuracy and efficiency. Similar to spatial carrier frequency interferometry, the spatial carrier frequency is also introduced in the assistant channel in the proposed method; therefore, the interferograms of this channel cannot be used to accurately extract the measured phase. However, in the DCI method, the assistant channel is only used to obtain the difference in the coefficients of the tilt phase plane relative to the first interferogram, thus avoiding the influence of the retrace error. The new concept of dual-channel information extraction in the DCI helps eliminate the calibration process of the retrace error before measurement, making this a calibration-free measurement.
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Furthermore, we propose a new algorithm, namely the stepwise local up-sampling phase correlation algorithm, to extract the coefficients of the tilt phase plane directly from the interferogram spectrum of the assistant channel. This eliminates the process of solving the tilt phase using the Fourier method and phase unwrapping. The remainder of this paper is organised as follows. In Section 2, the principle of the DCI is described in detail. The DCI phase extraction technique is introduced in Section 3. The results of simulations performed using the proposed dual-channel phase extraction algorithm are reported in Section 4. The results of experiments conducted to verify the vibration resistance performance of the DCI, including a comparison with spatial phase-shifting interferometry (SPSI), are presented in Section 5. The obtained results are discussed in Sections 6 and 7 concludes the study. 2. Optical principle of a dual-channel interferometer Fig. 1 shows the schematic of the proposed DCI. The interference system is divided into three parts: a light source and main- and assistantchannel interference optical paths. The light source comprises a laser, collimating lenses L1 and L2, and a half-wave plate (HWP). Fig. 2(a) shows the schematic of the main channel. After passing through a polarisation beam splitter (PBS), the laser beam can be divided into transmitted and reflected beams, i.e. p and s waves, respectively. After passing through a 𝜆/4 wave plate (QWP1), the s wave is reflected by a reference flat (RF1), after which it passes through QWP1 again, the fastaxis of which is at an angle of 45° with the s wave. The polarisation direction of the s wave is rotated by 90° after passing through QWP1 twice, and this wave becomes the reference beam of the main channel. The p wave transmitted from the PBS passes through another 𝜆/4 wave plate (QWP2) and divergence (L3), subsequently passing through QWP2 again after being reflected by a test sphere (TS). The fast-axis of QWP2 is at an angle of 22.5° with the p wave. After passing through QWP2 twice, the polarisation direction of the p wave is rotated by 90°; this wave becomes the test beam. The beam is then split by the PBS into a transmitted p wave and a reflected s wave, and the reflected beam becomes the test beam of the main channel. The test and reference beams of the main channel are orthogonal and linearly polarised and are synthesised using a polaroid (P3) to form an interferogram on camera (D) after passing through a mirror (RM) and objective lenses L4 and L5. Fig. 2(b) shows the schematic of the assistant channel. The reflected beam from the beam splitter (BS1) passes through a polaroid (P1) and a standard reference flat (RF2), becoming the reference beam of the assistant channel. The p wave returning from the PBS of the main channel is the test beam of the assistant channel. After being synthesised by the polaroid (P2), the reference and test beams pass through the objective lenses L4 and L5 to form an interferogram on D. In the DCI, the reference flats, RF1 and RF2, are mounted in one frame to ensure synchronous vibration of the two channels. A large carrier frequency is introduced in the assistant channel by separately adjusting the tilt angles of RF1 and RF2, while that of the main channel remains small. During testing, camera D simultaneously acquires a series of interferograms of the two channels. The interferogram sequence of the assistant channel is used for solving the coefficients of the vibration tilt phase plane, whereas the interferogram sequence of the main channel is used for solving the measured phase distribution by combining the tilt phase plane obtained from the assistant channel. The process of solving the two-channel phase is given in Section 3. 3. Phase extraction algorithm In the presence of vibration, the interference intensity of the main channel is as follows: 𝐼𝑛 (𝑥, 𝑦) = 𝑎(𝑥, 𝑦) + 𝑏(𝑥, 𝑦) cos(𝜑(𝑥, 𝑦) + 𝑃𝑛 (𝑥, 𝑦)),
(1)
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Fig. 1. Schematic of a dual-channel interferometer (DCI).
Fig. 2. Flat schematics of the DCI: Optical path of the (a) main channel and (b) assistant channel.
where𝑛 = 1, 2, ⋅ ⋅ ⋅, 𝑁, a(x, y)is the background intensity, b(x, y) is the modulation amplitude, 𝜑(x, y) is the phase map, and Pn (x, y) is the vibration tilt phase plane. Pn (x, y) is defined as follows: 𝑃𝑛 (𝑥, 𝑦) = 𝛼𝑛 𝑥 + 𝛽𝑛 𝑦 + 𝛾𝑛 .
(2)
According to the structural characteristics of the optical system described in Section 2, RF1 and RF2 are integral, and there is a tilt angle between them. Therefore, the tilt phase planes of the main and assistant channels are the same. Thus, the interference intensity expression for the assistant channel is as follows: 𝐼𝑛′ (𝑥, 𝑦) = 𝑎′ (𝑥, 𝑦) + 𝑏′ (𝑥, 𝑦) cos(𝜑(𝑥, 𝑦) + 𝐸(𝑥, 𝑦) + 𝑃𝑛 (𝑥, 𝑦)),
obtaining Pn (x, y). The measured phase 𝜑(x, y) can then be obtained from the main-channel fringe patterns using the main channel phase extraction algorithm described in Section 3.2.
(3)
Here, E(x, y) is the carrier-frequency phase and retrace error introduced by the tilt of RF2. In Eqs. (1) and (3), we assume that 𝑃𝑛 (𝑥, 𝑦) = 0. To solve the phase map 𝜑(x, y), the assistant-channel fringe patterns are first processed using the tilt phase plane coefficient extraction algorithm, described in Section 3.1, to determine 𝛼 n , 𝛽 n , and 𝛾 n , thereby
3.1. Extraction algorithm for tilt phase plane coefficients To obtain the tilt phase plane, we propose a tilt phase plane coefficient extraction algorithm. In this method, a Fourier transform is performed on the dense fringe pattern of the assistant channel, and the carrier-frequency coefficients (defined as kxn , kyn , andkn ) of each fringe pattern are directly calculated in the frequency domain. The tilt phase plane coefficients 𝛼 n ,𝛽 n , and 𝛾 n can be obtained from kxn , kyn , andkn . The proposed algorithm can extract the tilt phase plane coefficients quickly and accurately without having to solve the phase of the interferogram, thus eliminating the process of solving the phase using the
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Fourier method and phase unwrapping and avoiding the Gibbs effect. This significantly improves the efficiency and accuracy of measurement. Eq. (3) can be rewritten in a complex form as follows: 𝐼′
𝑛 (𝑥, 𝑦)
𝑎′ (𝑥, 𝑦) +
= exp(𝑗(𝑘𝑥𝑛 𝑥 + 𝑘𝑦𝑛 𝑦 + 𝑘𝑛 ))𝑐(𝑥, 𝑦) + exp(−𝑗(𝑘𝑥𝑛 𝑥 + 𝑘𝑦𝑛 𝑦 + 𝑘𝑛 ))𝑐 ∗ (𝑥, 𝑦),
(4)
where kxn , kyn , and kn are the carrier-frequency coefficients of the assistant channel, 𝑐(𝑥, 𝑦) = 1∕2𝑏′ (𝑥, 𝑦) exp(𝑗 𝜑𝑛 (𝑥, 𝑦)), and c∗ (x, y) is the conjugate of c(x, y). We define: Γ(𝑐(𝑥, 𝑦)) = 𝐶(𝑥, 𝑦),
(5)
3.2. Main channel phase extraction algorithm The retrace error can be neglected because of the small number of fringes in the main-channel interferograms. Thus, the measured phase 𝜑(x, y) can be calculated by combining the tilt phase plane obtained from the assistant channel with the main-channel interferograms. Defining, 𝑐1 (𝑥, 𝑦) = 𝑎(𝑥, 𝑦) 𝑐2 (𝑥, 𝑦) = 𝑏(𝑥, 𝑦) cos(𝜑(𝑥, 𝑦)) , 𝑐3 (𝑥, 𝑦) = −𝑏(𝑥, 𝑦) sin(𝜑(𝑥, 𝑦))
(11)
where Γ( · ) is the Fourier transform operation. By performing a 2D Fourier transform on the interferogram of size MA × NA of the assistant channel, we can obtain the following.
and omitting (x, y) for convenience, Eq. (1) can be rewritten as follows:
𝐹 (𝑢, 𝑣) = Γ(𝐼 ′ 𝑛 (𝑥, 𝑦)) = 𝐴′ (𝑢, 𝑣) + exp(𝑗 𝑘𝑛 )𝐶(𝑢 + 𝑓𝑥 , 𝑣 + 𝑓𝑦 ) + exp(−𝑗 𝑘𝑛 )𝐶 ∗ (𝑢 − 𝑓𝑥 , 𝑣 − 𝑓𝑦 ),
We denote the measured interferograms as 𝐼𝑛′′ (𝑥, 𝑦). The sum of the squared differences between the measured and theoretical interferograms predicted using Eq. (12) is given by ∑𝑁 2 𝑆= (𝐼𝑛 − 𝐼 ′′ 𝑛 ) , (13)
(6)
where A′(u, v) is the spectrum of the background intensity, and fx and fy are the coordinates of the positive and negative first-order peaks in the spectrum, respectively: 𝑓𝑥 = 𝑘𝑥𝑛 𝑁𝐴 ∕2𝜋, 𝑓𝑦 = 𝑘𝑦𝑛 𝑀𝐴 ∕2𝜋.
(7)
To extract fx and fy quickly and accurately, we propose the stepwise local up-sampling phase correlation algorithm, which includes a local up-sampling phase correlation algorithm and a stepwise algorithm. This algorithm only applies up-sampling to the adjacent areas of the peak, thus reducing the computation time and saving memory. The local up-sampling phase correlation algorithm implements Fourier transform on partial regions through matrix Fourier transform. The matrix Fourier transform of a 2D discrete signal 𝐼𝑛′ (𝑋, 𝑌 ) can be expressed as follows: ( ) ( ) 𝐹 (𝑈 , 𝑉 ) = exp −2𝜋jU𝑋 𝑇 𝐼𝑛′ (𝑋, 𝑌 ) exp 2𝜋jY𝑉 𝑇 ,
(8)
where 𝑋 = (𝑥0 , 𝑥1 , ⋅ ⋅ ⋅, 𝑥𝑁𝐴 −1 )𝑇 , 𝑌 = (𝑦0 , 𝑦1 , ⋅ ⋅ ⋅, 𝑦𝑀𝐴 −1 )𝑇 , 𝑈= (𝑢0 , 𝑢1 , ⋅ ⋅ ⋅, 𝑢𝑁𝐵 −1 )𝑇 , and 𝑉 = (𝑣0 , 𝑣1 , ⋅ ⋅ ⋅, 𝑣𝑁𝐵 −1 )𝑇 . We first extract the approximate position (fx0 , fy0 ) of the peak using the conventional method and then give the up-sampling area size m(1 ≤ m ≤ 2) and sampling magnification r. The sampling step is 1/r, and the number of sampling points is 𝑁𝐵 = 𝑚𝑟. i indicates the ordinal of the sampling points; accordingly, 𝑢𝑖 = 𝑓𝑥0 − 𝑁𝐵 ∕2 + 𝑖 and 𝑣𝑖 = 𝑓𝑦0 − 𝑁𝐵 ∕2 + 𝑖. The up sampling to the adjacent areas of the peak can be achieved using Eq. (8). When the required positioning accuracy is high, it is necessary to increase the sampling magnification r. To improve the computation speed and save computation memory, it is necessary to use the stepwise algorithm. For example, when 𝑟 = 10000, step 1: take 𝑟1 = 10 and perform the first up sampling according to the above method to locate the peak position of pixel/10 precision; step 2: perform the up sampling again with magnification r1 on the data sampled in step 1 to locate the peak position of pixel/100 precision; step 3: Repeat the above steps until 10,000 times sampling and locate the high-precision peak positions fx and fy . kxn and kyn can then be obtained using Eq. (7). After determining fx and fy , kn can be obtained using Eqs. (6) and (8): ( ( )) ( ( )) 𝑘𝑛 = 𝐴𝑟𝑔 𝐹 𝑓𝑥 , 𝑓𝑦 = 𝐴𝑟𝑔 exp −𝑗𝑘𝑛 ( ( ) ( )) = 𝐴𝑟𝑔 exp −2𝜋𝑗 𝑓𝑥 𝑋 𝑇 𝐼 ′ 𝑛 (𝑋, 𝑌 ) exp 2𝜋𝑗𝑌 𝑓𝑦 ,
(9)
where Arg( · ) is the function used to calculate the phase angle. After obtaining the carrier-frequency coefficients of the assistantchannel interferograms using the above algorithm according to the equation 𝛼𝑛 = 𝑘𝑥𝑛 − 𝑘𝑥1 , 𝛽𝑛 = 𝑘𝑦𝑛 − 𝑘𝑦1 , 𝛾𝑛 = 𝑘𝑛 − 𝑘1 ,
𝐼𝑛 = 𝑐1 + 𝑐2 cos(𝑃𝑛 ) + 𝑐3 sin(𝑃𝑛 ).
(12)
𝑛=1
where N is the total number of interferograms, and c1 , c2 , and c3 are three unknowns. The unknown variables can be determined by minimising S using the least-squares criterion: 𝜕𝑆 ∕𝜕 𝑐1 = 0, 𝜕𝑆 ∕𝜕 𝑐2 = 0, 𝜕𝑆 ∕𝜕 𝑐3 = 0.
(14)
The solutions to the above three equations are given by the following linear matrix equations: 𝐶 = 𝐴−1 𝐵, where [ 𝐶 = 𝑐1 𝐵=
(15)
𝑐2
[ ∑ 𝑁
𝑛=1
]T
𝑐3
𝐼 ′′ 𝑛
∑𝑁
𝑁 ⎡ ∑ 𝐴=⎢ 𝑁 cos(𝑃𝑛 ) ⎢ ∑𝑛𝑁=1 ⎣ 𝑛=1 sin(𝑃𝑛 )
,
𝑛=1
(16) 𝐼 ′′ 𝑛 cos(𝑃𝑛 )
∑𝑁
𝑛=1
∑𝑁 cos(𝑃 ) ∑𝑁𝑛=1 2 𝑛 cos (𝑃𝑛 ) ∑𝑁 𝑛=1 ′ 𝑛=1 sin(𝛿 𝑛 ) cos(𝑃𝑛 )
𝐼 ′′ 𝑛 sin(𝑃𝑛 )
]𝑇
,
(17)
∑𝑁 sin(𝑃 ) ⎤ ∑𝑁 𝑛=1 ′ 𝑛 ⎥ cos ( 𝛿 𝑛 ) sin(𝑃𝑛 )⎥. (18) 𝑛=1 ∑𝑁 2 ⎦ 𝑛=1 sin (𝑃𝑛 )
After determining c1 and c2 , we can obtain the phase map, i.e. 𝜑(x, y), as follows: 𝜑 = tan−1 (−𝑐3 ∕𝑐2 ).
(19)
4. Simulations To verify the feasibility and accuracy of the DCI phase extraction algorithm, we carried out simulation experiments. In the experiment, the test phase is fitted using the Zernike function. Table 1 lists the fitting coefficients. According to the tilt plane coefficients, listed in Table 2, eight groups of the main-channel and assistant-channel interferograms are simulated, and the resolution of the interferogram is 512 × 512. Fig. 3(a)–3(g) show the main-channel interferograms. Fig. 3(h) shows the assistant-channel interferogram with a high carrier frequency. The coefficients of the tilt phase planes can be obtained using the proposed tilt phase plane coefficient extraction algorithm. We set the sampling magnification 𝑟 = 10000; therefore, the location precision of the peak is pixel/10,000. Table 2 lists the coefficients. The maximum errors of the calculated tilt plane coefficients 𝛼 n , 𝛽 n , and 𝛾 n with respect to the real value are 5 × 10−6 rad/pixel, 3 × 10−6 rad/pixel, and 0.0255 rad. Table 1 Zernike Coefficients of the test phase for the simulation.
(10)
we can obtain the tilt phase plane generated using the coefficients 𝛼 n , 𝛽 n , and𝛾 n .
Zernike Terms
1
2
3
4
Coefficients
0.03
0.02
0.15
0.30
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Optics and Lasers in Engineering 127 (2020) 105981
Table 2 Extracted results of tilt phase plane. Frame
2 3 4 5 6 7 8
𝛼 (×10−2 rad/pixel)
𝛽 (×10−2 rad/pixel)
𝛾 (rad)
Real
Calculated
Real
Calculated
Real
Calculated
−2.0783 −0.4423 −1.5128 −1.2635 −1.8058 −0.5167 −1.1404
−2.0786 −0.4423 −1.5129 −1.2639 −1.8059 −0.5175 −1.1409
−0.6727 1.2799 0.0754 0.6520 2.0253 −0.2166 −0.2048
−0.6727 1.2800 0.0753 0.6518 2.0256 −0.2167 −0.2046
1.6514 4.6139 3.2902 2.5300 5.9449 1.4650 4.4651
1.6478 4.6091 3.3177 2.5555 5.9456 1.4634 4.4452
This proves the high precision of the proposed local up-sampling phase correlation algorithm. Fig. 4(a) and 4(b) show the real test phase and calculated phase extracted using the proposed method: the peak-tovalley (PV) values are 0.5977𝜆 and 0.5980𝜆, respectively, and the root mean square (RMS) values are 0.126490𝜆 and 0.126497𝜆, respectively. Fig. 4(c) shows the residual error between the results shown in Figs. 4(a) and 4(b): PV = 0.003𝜆 and RMS = 5.5325 × 10−4 𝜆. The simulation results prove that the proposed DCI phase extraction algorithm is feasible and accurate. 5. Experiments and analysis 5.1. Experimental setup
Fig. 3. Interferograms obtained from simulation.
To verify the feasibility of the DCI, we constructed an experimental setup based on the optical path principle. The light-source includes an HeNe laser with a wavelength of 𝜆 = 632.8 nm. The divergence, L3, has an F number of 5 and a focal length of 50 mm. The optical element to
Fig. 4. Simulation results: (a) real phase, (b) calculated phase, (c) residual error.
M. Duan, Y. Zong and Y. Xu et al.
Optics and Lasers in Engineering 127 (2020) 105981
Fig. 5. Results of the comparative experimental: (a) four phase-shifting interferograms, (b) phase obtained by four-step phase-shifting interferometry.
be measured is a spherical mirror having a diameter of 25 mm and a curvature radius of 125 mm. The focal lengths of the imaging lenses L2 and L3 are 150 and 75 mm, respectively. The detector is a CMOS camera (Point Grey, GS3-U3-23S6M-C) with a resolution of 1920 × 1200 and a pixel size of 5.8 𝜇m. Two standard mirrors are attached onto the same substrate to ensure synchronous vibration. In the following experiment, the ambient temperature is 23 ◦ C, and the vibration frequency is in the range of 20–200 Hz. The experimental environment is generally representative.
5.2. Experiments in vibrating environment To verify the performance of the DCI with respect to environmental vibration resistance, we conducted experiments in a vibrating environment. In the vibrating environment, 20-frame interferograms were acquired. The number and direction of the fringes varied considerably. Firstly, Using the four-step phase-shifting interferometry, we calculated the measured phase from the first four interferograms of the main channel. Figs. 5(a) and (b) show the four phaseshifting interferograms and calculation results. It can be seen from Fig. 5(b) that there are obvious ripple errors caused by vibration, with PV = 0.8109𝜆 and RMS = 0.0827𝜆. And then using the previously mentioned algorithm, we obtained the measured phase from the acquired interferograms. Figs. 6(a) and (b) show the three-frame interferograms of the main and assistant channels. The measured phase was extracted using the DCI phase extraction algorithm. Fig. 6(c) and 6(d) show the results: PV = 0.3471𝜆 and RMS = 0.0578𝜆. The experimental results show that the phase extracted by DCI method has no ripple error, despite performing the experiment in a vibrating environment, thus proving the excellent vibration-resistance performance of the DCI. To verify the reliability of the DCI measurements, we conducted comparative experiments with SPSI. To simultaneously perform DCI and spatial phase-shifting measurements in the same optical path, we replaced the polarising plate P3 in the main channel with a 𝜆/4 wave plate QWP3 and the camera with a polarisation camera (FLIR, BFS-U351S5P-C, number of pixels: 2448 × 2048). In this experimental setup, only two devices were replaced. Thus, the system errors and aberration
in the two schemes were the same during the experiment, and the experimental results were more comparable. The polarisation camera captures 20-frame fringe patterns, each of which can be divided into four interferograms with a phase shift of 𝜋/2. For the DCI method, we divided all the fringe patterns into four interferograms and then calculated the measured phase using the interferograms with the same phase shift. Fig. 8(a) shows the phase results: PV = 0.3321𝜆 and RMS = 0.0599𝜆; For the SPSI method, we divided one fringe pattern into four interferograms and extracted the region of the main channel. Fig. 7 shows the interferograms. Fig. 8(b) shows the phase distribution calculated using the four-step phase-shifting method: PV = 0.3373𝜆 and RMS = 0.0595𝜆. Fig. 8(c) shows the residual error between the phase extracted using the DCI and SPSI: PV = 0.1002𝜆 and RMS = 0.0093𝜆. The phase results of the two methods are consistent. A small ripple error equal to the frequency of the fringe patterns can still be seen in the residual error phase, because of the inconsistent background intensity of the four spatial phase-shifting interferograms. Thus, the experimental results demonstrate the reliability of the DCI measurements. 6. Discussion The experiments conducted in the study verify the effectiveness and reliability of the DCI with respect to vibration resistance. Considering the optical path, system devices, and experimental results, the DCI can be beneficial and practical for optical testing in vibration environments. (i) Similar to the spatial carrier frequency interferometry, a high carrier frequency is also introduced in the assistant channel in the DCI method, thus generating a large non-common optical path retrace error and a systematic error in this channel. Therefore, the interferograms of this channel cannot be used to extract the final phase. However, the assistant channel is only used to obtain the difference in the coefficients of the tilt phase plane relative to the first interferogram. Moreover, as the vibration of the two channels is synchronous, the tilt phase plane generated based on the coefficients of the assistant channel can reflect the vibration of the main channel. Finally, the measured phase is extracted using the interferograms of the main channel and the tilt phase plane, with no influence of the retrace
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Optics and Lasers in Engineering 127 (2020) 105981
Fig. 6. Results of the vibration-resistant experiments: (a) interferograms of the main channel, (b) interferograms of the assistant channel, (c) 2D and 3D phase maps.
Fig. 7. Spatial phase-shifting interferograms extracted by a polarization camera.
error. In order to reduce the non-common optical path retrace error of the main channel and obtain more accurate coefficients of the tilt phase plane from the assistant channel, the number of fringe of the two channels should be within 5 and between 100 and 200 respectively. The new concept of dual-channel information extraction in the DCI helped eliminates the calibration process of retrace error before measurement, thus simplifying the measurement process and making it easier to apply the system to actual measurements.
(ii) Compared with the PDI method proposed by our research group recently, the DCI method has some advantages. The role of the carrier frequency pass in the PDI is taken up by the assistant channel in the DCI; this simplifies the optical system and operation process. The reference and test beams of the assistant channel have a smaller lateral deviation on the detector target surface when a high carrier frequency is introduced. Thus, the spatial resolution of the detector can be better utilized. Furthermore, the proposed stepwise local up-sampling phase correlation algorithm can extract the coefficients of the tilt phase plane directly from the interferogram spectrum of the assistant channel. This eliminates the process of solving the tilt phase using the Fourier method and phase unwrapping and avoids the Gibbs effect, thus significantly improving the efficiency and accuracy of measurement. There are also some disadvantages for the DCI compared with the PDI. In the DCI, the reference beams of the main channel and assistant channel are generated by the two reference flats respectively. Although the two reference flats are mounted in the same structure, the structural stability will still decrease after
M. Duan, Y. Zong and Y. Xu et al.
Optics and Lasers in Engineering 127 (2020) 105981
Fig. 8. Results of the comparative experiments: (a) phase map extracted using the DCI, (b) phase map extracted using the SPSI, (c) residual error.
long-term use, which will affect the vibration synchronicity of the two channels and cause errors in measurement results. (iii) The DCI efficiently uses two interference channels to directly determine the vibration tilt phase plane; this is a novel technique in interferometry. Although the two channels are both off-axis imaged, the beams of them are collimated and normal incidence on the camera and symmetrical about the optical axis. So off-axis imaging will not affect the test results. Environmental vibration was utilized to perform ‘phase shifting’ for the surface measurements of optical components. The DCI measurement method not only eliminates the need for a vibration isolation platform, but also the need for expensive and precise phase-shifting devices. Moreover, as the DCI is of the temporal phase-shifting type, the inconsistency in the contrast between the fringe patterns due to polarisation elements is avoided. (iv) The DCI method has some limitations. Similar to most beam splitting interferometry methods, the proposed DCI method requires calibration between the interferograms of the main and assistant channels. Therefore, the calibration process affects the measurement results.
of the DCI, including a comparison with SPSI. The experimental results demonstrated that the DCI has good vibration resistance performance. Compared with the standard vibration-resistant interferometer, the DCI uses environmental vibration for ‘phase shifting’, does not require a vibration isolation platform or precise phase shifting devices, and has advantages such as structural simplicity, ease of operation, and low cost. Funding National Natural Science Foundation of China (NSFC) (61,975,079), (61,475,072); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0399). Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
7. Conclusions
Supplementary materials
In summary, we proposed a DCI for vibration-resistant optical testing. The DCI employs a dual-channel interference optical path design. An assistant channel is used for calculating the coefficients of the tilt phase plane using the stepwise local up-sampling phase correlation algorithm; the main channel fringe patterns are combined with the tilt phase plane generated based on the coefficients of the assistant channel to obtain the measured phase using the least-squares method. Simulations were performed using the DCI phase extraction algorithm, and experiments were conducted to verify the vibration-resistant performance
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.optlaseng.2019.105981. CRediT authorship contribution statement Mingliang Duan: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Writing - review & editing, Formal analysis. Yi Zong: Software, Formal analysis, Investigation. Yixuan Xu: Resources, Data curation. Guolian Chen: Software,
M. Duan, Y. Zong and Y. Xu et al.
Data curation. Wenqian Lu: Software, Data curation. Cong Wei: Software, Resources. Rihong Zhu: Supervision, Project administration. Lei Chen: Resources, Project administration. Jianxin Li: Conceptualization, Methodology, Resources, Supervision, Project administration. References [1] Malacara D, Servín M, Malacara Z. Interferogram analysis for optical testing. 2nd ed. Taylor and Francis; 2005. [2] Malacara D. Optical shop testing. 3rd ed. CRC Press; 2007. [3] de Groot PJ. Vibration in phase-shifting interferometry. J Opt Soc Am A 1995;12(2):354–65. [4] Deck LL. Suppressing phase errors from vibration in phase-shifting interferometry. Appl Opt 2009;48(20):3948–60. [5] Deck LL. Model-based phase shifting interferometry. Appl Opt 2014;53(21):4628–36. [6] de Groot PJ, Deck LL. Numerical simulations of vibration in phase-shifting interferometry. Appl Opt 1996;35(13):2172–8. [7] Okada K, Sato A, Tsujiuchi J. Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry. Opt Commun 1991;84(3–4):118–24. [8] Wang Z, Han B. Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations. Opt Laser Eng 2007;45(2):274–80. [9] Cai LZ, Liu Q, Yang XL. Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps. Opt Lett 2003;28(19):1808–10. [10] Xu XF, Cai LZ, Meng XF, Dong GY, Shen XX. Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry. Opt Lett 2006;31(13):1966–8. [11] Xu J, Jin W, Chai L, Xu Q. Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method. Opt Express 2011;19(21):20483–92. [12] Vargas J, Sorzano COS, Estrada JC, Carazo JM. Generalization of the principal component analysis algorithm for interferometry. Opt Commun 2013;286:130–4. [13] Vargas J, Sorzano COS. Quadrature component analysis for interferometry. Opt Laser Eng 2013;51:637–41. [14] Liu F, Wu Y, Wu F. Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology. Opt Express 2015;23(8):10794–807. [15] Tay CJ, Quan C, Chen L. Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift. Appl Opt 2005;44(8):1401–9.
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