- Email: [email protected]
Advanced simultaneous phase-shifting Fizeau interferometer
Optics and Laser Technology 111 (2019) 134–139
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Advanced simultaneous phase-shifting Fizeau interferometer a
a,⁎
a
a
a
a
b
Wenhua Zhu , Lei Chen , Ying Yang , Rui Zhang , Donghui Zheng , Zhigang Han , Jinpeng Li a b
T
School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China Nanjing Astronomical Instruments Company Limited, Chinese Academy of Sciences, Nanjing 210042, China
HIGHLIGHTS
advanced simultaneous phase-shifting Fizeau interferometer. • An imaging resolution and measurement accuraccy. • High • Optical zoom is achieved in measuring. ARTICLE INFO
ABSTRACT
Keywords: Simultaneous phase-shifting Fizeau interferometer Single shot Dynamic measurement
This study proposes an advanced simultaneous phase-shifting Fizeau interferometer. The proposed interferometer introduces a beam expander into the simultaneous phase-shifting Fizeau interferometer which employs four point sources to produce four phase shifting interferograms on a detector simutaneously. The beam expander is used to increase the aperture of the lens array and to decrease the inclination of the beam incident on the test surface such that the imaging resolution and the measurement accuracy of the interferometer can be improved. In addition, the optical zoom can be achieved by changing the magnification of the beam expander. A step plate was experimentally measured using the proposed interferometer. The phase step is clearly imaged in the detector, and the measured results are in good agreement with those obtained by the temporal phase-shifting Fizeau interferometer, which indicates that the proposed interferometer is suitable for normal apertures.
1. Introduction Fizeau interferometers are widely used for optical measurements because of their common-path configuration, where the aberrations of the optical system can be removed from the measured results [1]. However, these Fizeau interferometers are based on temporal phase shifting interferometry, which enables them to extract phase information such that the measurement results are inaccurate in the presence of unstable conditions such as vibration, air turbulence, or when the object under test is in motion [2]. In recent years, there has been an increasing interest in dynamic measurements especially in astronomical optics and high-power laser fields [3,4]. In order to apply Fizeau interferometers to dynamic measurements, several methods have been proposed [5–7]. However, many of them break the common-path configuration of the Fizeau interferometer, thereby introducing retrace errors. For example, Sykora and Groot realized dynamic measurement by tilting the reference mirror to introduce a high linear-carrier frequency to the interferogram [8]; Szwaykowski et al. captured three phase-shifting interferograms
⁎
simultaneously using three charge-coupled devices (CCDs), and the reference and test wavefronts were separated [9]; Chatterjee and Kumar proposed a cyclic path optical configuration with the reference and test beam couples tilted mutually such that they realized dynamic measurement in the Fizeau interferometer [10]. In order to increase the measurement accuracy, the retrace errors must be avoided. Abdelsalam et al. presented a method that replaces the reference mirror with a quarter waveplate [11]. This configuration appears to be perfect in terms of performing dynamic Fizeau interferometry, but it is difficult to fabricate a quarter waveplate with high flatness, especially in large apertures. Kimbrough et al. designed a low-coherence polarized light source [12]. Two couples of reference and test beams are superimposed on the CCD target, but only one pair of beams is coherent, and then the fringe contrast is lowered. Recently, we proposed a dynamic Fizeau interferometry, where the simultaneous phase shifting is achieved by changing the positions of four point sources [13]. While the point sources are not required to be polarized, the reference and test beams share a common path. This method exhibits good performance in a large aperture Fizeau interferometer, but for normal apertures such as
Corresponding author. E-mail address: [email protected] (L. Chen).
https://doi.org/10.1016/j.optlastec.2018.09.040 Received 3 April 2018; Received in revised form 9 August 2018; Accepted 18 September 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
Fig. 1. Optical layout of the proposed interferometer.
Fig. 2. Principle of phase shifting.
those with 100-mm size, there will be a decrease in the imaging resolution and measurement accuracy. The main causes for these limitations are the aperture of the image lens array and the aberrations of the collimated wavefront in the proposed Fizeau interferometer. In this study, we present the analysis for these decreases and propose an advanced simultaneous phase-shifting Fizeau interferometer, where the decreases can be avoided such that the Fizeau interferometer with normal aperture can achieve as good a performance as that with a large aperture. In addition, the advanced configuration can realize optical zoom in measuring.
2. Theory 2.1. Optical layout The optical layout of the advanced simultaneous phase-shifting Fizeau interferometer is shown in Fig. 1. The spherical wavefront emitted from the fiber laser is collimated by a collimator L1 before being transmitted through a phase grating to form multiple plane wavefronts. The wavefronts are then converged to a stop mask by a lens L2. Only the ( ±1, ± 1) orders are selected as the point-source array, which
135
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
Fig. 3. Optical layout of the previous configuration.
have identical intensity owing to the phase modulation of the chessboard grating, with the height of the phase step equal to π for the central wavelength of the spectral band [14]. Four spherical wavefronts emitted from the point-source array are collimated by a collimator L3 and then transmitted through the beam-splitting film before being expanded by a beam expander that comprises a negative lens L4 and a positive lens L5. Each expanded beam is reflected by the reference and test mirrors, and forms a pair of coherent beams. All of the coherent beams are returned to the imaging system, where the lens L6 determines the intervals of the four pairs of coherent beams at its focal plane, and the combination of the lens array L7 and lens L8 determines that four phase-shifting interferograms are imaged clearly in different positions of the detector, thereby realizing dynamic measurement.
the phase step between every two adjacent interferograms arranged in a special sequence is π/2, where l is the interval of two adjacent point sources. In order to realize dynamic measurements, four phase-shifting interferograms should be imaged clearly at different positions of the detector such that the test mirror is conjugated to the detector, and then the distance between L8 and the detector can be derived as
lD =
Unlike the conventional Fizeau interferometer, the phase shifting of the proposed interferometer is achieved by changing the inclination of the beam incident on the reference mirror. The principle of phase shifting is shown in Fig. 2, where one of the point sources shown in Fig. 1 is used to present the optical path. The optical path difference between the reference and test wavefronts can be expressed as [13]
D < f5 +
2 D
2
(1)
(2)
y=
(4m + 1) f32 f52 , 8lDf42
m
Integer
(2n + 1) f32 f52 , 4lDf42
n
Integer
f52 (f6 + f4 ls ) f42
.
(5)
The entrance pupil of the imaging system determines the imaging resolution. In the previous configuration shown in Fig. 3 [13], the resolution is limited by the lens array L7 because its maximum aperture is equivalent to the interval of two adjacent point sources. This interval cannot be too large, or there will be large aberrations in the collimated wavefront, which introduces systematic errors. However, in the proposed configuration, the relationship between the aperture a of L7 and the interval l is such that a = lf6 / f3, which means that both large a and small l can be obtained by choosing a long focal length of L6 or a short focal length of L3, thereby improving the imaging resolution. The measurement accuracy is influenced by not only the reference surface but also the collimated wavefront distortion. The incident wavefront has an inclination with reference and test mirrors such that the reference wavefront has a displacement with the test wavefront. Then, the collimated wavefront distortion contained in the reference and test wavefronts cannot cancel each other out. Consider a pair of coherent wavefronts as an example. The phase difference between the reference
where is the wavelength, d is the offset from the point source to the front focus of L3, and f3 , f4 , and f5 are the focal lengths of L3, L4, and L5, respectively. Eq. (2) indicates that the phase shifting can be achieved by varying d . If the offsets of the point-source array shown in Fig. 1 are different, then simultaneous phase shifting can be obtained. In particular, when the offset from the center of point-source array to the focus along x and y directions meet
x=
(4)
2.3. Imaging resolution and measurement accuracy
d 2f4 2 f32 f5 2
,
Because L4 and L5 comprise the beam expander, f4 is short. We have f4 . Thus, Eq. (5) can be satisfied easily by giving f52 f42 and f6 f6 + f4 ls > 0 .
where D is the cavity length between the reference and test mirrors and is the inclination of the beam incident on the reference mirror, which is determined by the beam expander, which is composed of L4 and L5 f4 / f5 . Becasue f3 d , = d/ f3 . Therefore, the phase such that = shift can be written according to Eq. (1) as
=
f52 f62 f7
where ls is the distance between L4 and L6, and the distance between L5 and the reference mirror is omitted; f6 , f7 , and f8 are the focal lengths of L6, L7, and L8, respectively, where L7 is placed at the rear focal plane of L6 and the front focal plane of L8. The condition that lD > f8 f82 / f7 should be satisfied to ensure that the four phase-shifting interferograms are non-overlapped according to the imaging analysis of Ref. [13]. Hence, based on Eq. (4)
2.2. Principle of simultaneous phase shifting
= 2D cos
f8 {f42 f5 f7 f8 Df42 f7 f8 + f52 [f62 (f7 f8 ) + f6 f7 f8 + f7 f8 (f4 ls )]}
, (3) 136
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
and test wavefronts can be expressed as
(p ) = =
T
T
(p ) +
(p ) +
+
2
+ 8
accuracy may be further improved by replacing it with a better reference mirror.
W (p )
D W (p ) s
,
4. Discussion
(6)
4.1. Design of the proposed interferometer
where p = (x , y ) is a spatial variable with Cartesian components x and y , T (p ) is the phase under test, W (p ) is the difference wavefront of the collimated wavefront, is the diameter of L5, and W (p )/ s is the partial derivative along the displacement direction. Eq. (6) indicates that the influence of the collimated wavefront distortion can be suppressed by increasing or decreasing . In the previous configuration [13], these conditions can be satisfied in a large aperture interf4 /f5 is referometer. However, in the proposed configuration, = duced with a beam expander such that a normal aperture interferometer can be applied. In addition, from Eq. (5),
8
f5 +
f52 (f6 + f4 f 42
If the beam expander is removed, the proposed interferometer is still a simultaneous phase-shifting Fizeau interferometer, but with a small aperture. Therefore, the design of the small-aperture interferometer should be optimized. According to Eq. (6), the influence of the collimated wavefront distortion can be suppressed by decreasing . This condition can be realized by increasing the focal length of L3. In additon, it is easier to control the collimated wavefront distortion by selecting a collimator with a long focal length. However, the focal length should not be too long; otherwise, the optical system will be extremely large because the relationship between the aperture a of L7 and the point-source array interval l is such that a = lf6 / f3. With respect to the designation of the beam expander, the magnification determines the measurement aperture of the proposed interferometer, which indicates that an optical zoom can be realized by varying the magnification. In addition, the larger the magnification, the higher the measurement accuracy.
ls )
max
W (p ) s
<
e,
(7)
which guarantees a high measurement accuracy, where max[·] is defined as the maximum and e is the maximum tolerable phase error. 3. Experiment
4.2. Imaging resolution
The experimental set up of the proposed interferometer system is depicted in Fig. 1. A He-Ne gas laser with a 632.8-nm wavelength was coupled in a fiber used as a point source. In order to produce a pointsource array with a small interval, the focal lengths of L1 and L2 are 100 mm and 60 mm, respectively, and a grating with a 38-μm pitch was designed to constrain the interval to be 2 mm. Then, the wavefronts emitted from the point-source array are transmitted through a collimating lens with a focal length of 200 mm before being expanded by a beam expander, where the focal lengths of L4 and L5 are −111.7 mm and 435.7 mm, respectively, and the expanded aperture is 80 mm. To guarantee measurement accuracy, an optical flat with a surface quality better than /20 was used as the reference mirror. A step plate with a diameter of 75 mm was placed 150 mm behind the reference mirror as the test mirror. The reference and test wavefronts reflected by the reference and test mirrors are returned to the imaging system, where a focal length of L6 was designed to be 300 mm such that the aperture of L7 can be increased from 2 mm to 3 mm, thereby improving the imaging resolution. ls is 70 mm, which meets the requirement of Eq. (5). The focal lengths of L7 and L8 were designed to be −40 mm and 35 mm, respectively, such that four phase-shifting interferograms are clearly imaged in the detector, from which the captured interferogram with 1200 × 1200 pixels is shown in Fig. 4(a). To make each phase step be π/2, we adjust the offset such that x = 0.16 mm and y = 0.32 mm according to Eq. (3). The four interferograms shown in Fig. 4(a) were denoted as I1, I2 , I3, and I4 by the addition of the corresponding phase shifts (0, /2, , 3 /2 ). Fig. 4(b) is the phase extracted from Fig. 4(a) using the phase-shifting algorithm [15] and phase-unwrapping method [16]. The phase step can be seen clearly, and its average height (AvgHgt) is 132.3 nm. Fig. 4(d) shows the test result of the same step plate from a ZYGO GPI interferometer, which is a conventional temporal phase-shifting Fizeau interferometer, where AvgHgt = 132.06 nm. To further compare the measurement accuracy, the upper phase steps of both results from Fig. 4(b) and (d) were selected. Both surfaces are shown in Fig. 4(c) and (e), where the peakvalley (PV) values are 0.040λ and 0.042λ, respectively, and the root mean square (RMS) values are 0.005λ and 0.003λ, respectively. Compared with the ZYGO GPI interferometer, the proposed interferometer achieves differences of 0.24 nm (AvgHgt), 0.002λ (PV) and 0.002λ (RMS). As the surface quality of the reference mirror is close to /20 , and the measured result has reached this limitation, the measurement
To analyze the imaging resolution, we used Zemax to simulate the imaging system of the proposed interferometer. According to the experimental parameters, the simulated imaging system is shown in Fig. 5(a), which shows that the primary limitation of the imaging resolution is the aperture of L7. More specifically, Fig. 5(b) and (c) present the modulation transfer function (MTF) with 2-mm and 3-mm diameters of L7, respectively. The spatial frequency is linearly proportional to the diameter. 4.3. Cavity length According to Eq. (6), the longer the cavity length D , the greater will be the collimated wavefront distortion. However, if D is too short, the offset ( x , y ) will be very large based on Eq. (3), thereby increasing the collimated wavefront distortion. In order to find the most suitable D , the square of the maximum displacement of the point-source array is given as 2 d max = ( x + l/2)2 + ( y + l/2)2
=
5 2f 34 f 54
64l2D2f 44
+
l2 2
+
3 f32 f52 8Df 42
,
(8)
where (m , n) = (0, 0) . Eq. (8) has 5 2f34 f54 /64l2D2f 44 = l2/2 . Thus, we have
10 f32 f52
D=
8l 2f42
a
minimum
,
when
(9)
which is the best cavity length for the measurement. 4.4. Error analysis The error sources mainly include the intensity aberration of four phase-shifting interferograms, the tilt of the test mirror, the distortion of the imaging system, the pixel mismatches of four phase-shifting interferograms, all of which have been discussed in Ref. [13], and the aberration of the collimated wavefront, which will be analyzed here. According to Eq. (8), the error term shown in Eq. (6) can be rewritten as
We =
137
8
3f 2 f 2 D 5f 4 f 4 l 2D 2 + 3 5 2 + 3 2 54 2 2 8 f4 64l f 4
W (p ) s
,
(10)
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
Fig. 4. Experimental results: (a) the interferogram captured by the detector; (b) the phase extracted from (a); (c) the upper phase step of (b); (d) the test result of the same step plate from a ZYGO GPI interferometer; (e) the upper phase step of (d).
which indicates that We is an increasing function of D . Therefore, if the design of the proposed interferometer is not ideal, resulting in a large aberration to the collimated wavefront, the best approach is to reduce the cavity length even though is increased.
introduced into the previous simultaneous phase-shifting Fizeau interferometer, which can not only increase the imaging resolution, but also improve the measurement accuracy such that the proposed interferometer can be applied to normal aperture measurements. In addition, optical zoom can also be achieved at different measurement magnifications by the beam expander. In comparison, the proposed interferometer has a performance that is identical to that of the temporal phase-shifting Fizeau interferometer.
5. Conclusion This study presents an advanced simultaneous phase-shifting Fizeau interferometer. In the proposed configuration, a beam expander is
Fig. 5. Imaging system simulation: (a) the simulated imaging system; (b) the MTF with 2-mm diameter of L7; (c) the MTF with 3-mm diameter of L7. 138
Optics and Laser Technology 111 (2019) 134–139
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Acknowledgments
[7] R. Smythe, R. Moore, Instantaneous phase measuring interferometry, Opt. Eng. 23 (1984) 361–364. [8] D.M. Sykora, P. de Groot, Instantaneous measurement Fizeau interferometer with high spatial resolution, Proc. SPIE 8126 (2011) 50–58. [9] P. Szwaykowski, R.J. Castonguay, F.N. Bushroe, Simultaneous phase shifting module for use in interferometry, United States patent US 7483145, B2, 2009. [10] S. Chatterjee, Y.P. Kumar, Polarization phase shifting Fizeau interferometer with a cyclic path optical configuration, Opt. Eng. 48 (2009) 1–6. [11] D.G. Abdelsalam, B. Yao, P. Gao, J. Min, R. Guo, Single-shot parallel four-step phase shifting using on-axis Fizeau interferometry, Appl. Opt. 51 (2012) 4891–4895. [12] B. Kimbrough, J. Millerd, J. Wyant, J. Hayes, Low-coherence vibration insensitive Fizeau interferometer, SPIE 62920F (2006) 1–12. [13] W. Zhu, L. Chen, Y. Yang, R. Zhang, D. Zheng, Z. Han, J. Li, 600-mm aperture simultaneous phase-shifting Fizeau interferometer, Opt. Laser. Technol. 104 (2018) 26–32. [14] W. Zhu, J. Li, L. Chen, D. Zheng, Y. Yang, Z. Han, Systematic error analysis and correction in quadriwave lateral shearing interferometer, Opt. Commun. 380 (2016) 214–220. [15] Y. Surrel, Fringe analysis, Photomechanics, Springer, Berlin, Heidelberg, 2000, pp. 55–102. [16] M.A. Herráez, D.R. Burton, M.J. Lalor, M.A. Gdeisat, Fast two-dimensional phaseunwrapping algorithm based on sorting by reliability following a noncontinuous path, Appl. Opt. 41 (2002) 7437–7444.
This project was supported by the National Natural Science Foundation of China (U1731115, 61505082), the Natural Science Foundation of Jiangsu Province (SBK2015041354, BK20160154), the Fundamental Research Funds for the Central Universities (30916014112-003), and the China Scholarship Council. References [1] D. Malacara, Optical Shop Testing, John Wiley & Sons, 2007. [2] D.M. Sykora, M.L. Holmes, Dynamic measurements using a Fizeau interferometer, in: SPIE. 8082, 80821R-1-12, 2011. [3] T. Shirai, T.H. Barnes, T.G. Haskell, Adaptive wave-front correction by means of alloptical feedback interferometry, Opt. Lett. 25 (2000) 773–775. [4] C. Hernandez-Gomez, J.L. Collier, S.J. Hawkes, C.N. Danson, C.B. Edwards, D.A. Pepler, I.N. Ross, T.B. Winstone, Wave-front control of a large-aperture laser system by use of a static phase corrector, Appl. Opt. 39 (2000) 1954–1961. [5] K. Freischlad, R. Eng, J.B. Hadaway, Interferometer for testing in vibration environments, SPIE 4777 (2002) 311–322. [6] L.L. Deck, Environmentally friendly interferometry, SPIE 5532 (2004) 159–169.
139
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Advanced simultaneous phase-shifting Fizeau interferometer a
a,⁎
a
a
a
a
b
Wenhua Zhu , Lei Chen , Ying Yang , Rui Zhang , Donghui Zheng , Zhigang Han , Jinpeng Li a b
T
School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China Nanjing Astronomical Instruments Company Limited, Chinese Academy of Sciences, Nanjing 210042, China
HIGHLIGHTS
advanced simultaneous phase-shifting Fizeau interferometer. • An imaging resolution and measurement accuraccy. • High • Optical zoom is achieved in measuring. ARTICLE INFO
ABSTRACT
Keywords: Simultaneous phase-shifting Fizeau interferometer Single shot Dynamic measurement
This study proposes an advanced simultaneous phase-shifting Fizeau interferometer. The proposed interferometer introduces a beam expander into the simultaneous phase-shifting Fizeau interferometer which employs four point sources to produce four phase shifting interferograms on a detector simutaneously. The beam expander is used to increase the aperture of the lens array and to decrease the inclination of the beam incident on the test surface such that the imaging resolution and the measurement accuracy of the interferometer can be improved. In addition, the optical zoom can be achieved by changing the magnification of the beam expander. A step plate was experimentally measured using the proposed interferometer. The phase step is clearly imaged in the detector, and the measured results are in good agreement with those obtained by the temporal phase-shifting Fizeau interferometer, which indicates that the proposed interferometer is suitable for normal apertures.
1. Introduction Fizeau interferometers are widely used for optical measurements because of their common-path configuration, where the aberrations of the optical system can be removed from the measured results [1]. However, these Fizeau interferometers are based on temporal phase shifting interferometry, which enables them to extract phase information such that the measurement results are inaccurate in the presence of unstable conditions such as vibration, air turbulence, or when the object under test is in motion [2]. In recent years, there has been an increasing interest in dynamic measurements especially in astronomical optics and high-power laser fields [3,4]. In order to apply Fizeau interferometers to dynamic measurements, several methods have been proposed [5–7]. However, many of them break the common-path configuration of the Fizeau interferometer, thereby introducing retrace errors. For example, Sykora and Groot realized dynamic measurement by tilting the reference mirror to introduce a high linear-carrier frequency to the interferogram [8]; Szwaykowski et al. captured three phase-shifting interferograms
⁎
simultaneously using three charge-coupled devices (CCDs), and the reference and test wavefronts were separated [9]; Chatterjee and Kumar proposed a cyclic path optical configuration with the reference and test beam couples tilted mutually such that they realized dynamic measurement in the Fizeau interferometer [10]. In order to increase the measurement accuracy, the retrace errors must be avoided. Abdelsalam et al. presented a method that replaces the reference mirror with a quarter waveplate [11]. This configuration appears to be perfect in terms of performing dynamic Fizeau interferometry, but it is difficult to fabricate a quarter waveplate with high flatness, especially in large apertures. Kimbrough et al. designed a low-coherence polarized light source [12]. Two couples of reference and test beams are superimposed on the CCD target, but only one pair of beams is coherent, and then the fringe contrast is lowered. Recently, we proposed a dynamic Fizeau interferometry, where the simultaneous phase shifting is achieved by changing the positions of four point sources [13]. While the point sources are not required to be polarized, the reference and test beams share a common path. This method exhibits good performance in a large aperture Fizeau interferometer, but for normal apertures such as
Corresponding author. E-mail address: [email protected] (L. Chen).
https://doi.org/10.1016/j.optlastec.2018.09.040 Received 3 April 2018; Received in revised form 9 August 2018; Accepted 18 September 2018 0030-3992/ © 2018 Elsevier Ltd. All rights reserved.
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
Fig. 1. Optical layout of the proposed interferometer.
Fig. 2. Principle of phase shifting.
those with 100-mm size, there will be a decrease in the imaging resolution and measurement accuracy. The main causes for these limitations are the aperture of the image lens array and the aberrations of the collimated wavefront in the proposed Fizeau interferometer. In this study, we present the analysis for these decreases and propose an advanced simultaneous phase-shifting Fizeau interferometer, where the decreases can be avoided such that the Fizeau interferometer with normal aperture can achieve as good a performance as that with a large aperture. In addition, the advanced configuration can realize optical zoom in measuring.
2. Theory 2.1. Optical layout The optical layout of the advanced simultaneous phase-shifting Fizeau interferometer is shown in Fig. 1. The spherical wavefront emitted from the fiber laser is collimated by a collimator L1 before being transmitted through a phase grating to form multiple plane wavefronts. The wavefronts are then converged to a stop mask by a lens L2. Only the ( ±1, ± 1) orders are selected as the point-source array, which
135
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
Fig. 3. Optical layout of the previous configuration.
have identical intensity owing to the phase modulation of the chessboard grating, with the height of the phase step equal to π for the central wavelength of the spectral band [14]. Four spherical wavefronts emitted from the point-source array are collimated by a collimator L3 and then transmitted through the beam-splitting film before being expanded by a beam expander that comprises a negative lens L4 and a positive lens L5. Each expanded beam is reflected by the reference and test mirrors, and forms a pair of coherent beams. All of the coherent beams are returned to the imaging system, where the lens L6 determines the intervals of the four pairs of coherent beams at its focal plane, and the combination of the lens array L7 and lens L8 determines that four phase-shifting interferograms are imaged clearly in different positions of the detector, thereby realizing dynamic measurement.
the phase step between every two adjacent interferograms arranged in a special sequence is π/2, where l is the interval of two adjacent point sources. In order to realize dynamic measurements, four phase-shifting interferograms should be imaged clearly at different positions of the detector such that the test mirror is conjugated to the detector, and then the distance between L8 and the detector can be derived as
lD =
Unlike the conventional Fizeau interferometer, the phase shifting of the proposed interferometer is achieved by changing the inclination of the beam incident on the reference mirror. The principle of phase shifting is shown in Fig. 2, where one of the point sources shown in Fig. 1 is used to present the optical path. The optical path difference between the reference and test wavefronts can be expressed as [13]
D < f5 +
2 D
2
(1)
(2)
y=
(4m + 1) f32 f52 , 8lDf42
m
Integer
(2n + 1) f32 f52 , 4lDf42
n
Integer
f52 (f6 + f4 ls ) f42
.
(5)
The entrance pupil of the imaging system determines the imaging resolution. In the previous configuration shown in Fig. 3 [13], the resolution is limited by the lens array L7 because its maximum aperture is equivalent to the interval of two adjacent point sources. This interval cannot be too large, or there will be large aberrations in the collimated wavefront, which introduces systematic errors. However, in the proposed configuration, the relationship between the aperture a of L7 and the interval l is such that a = lf6 / f3, which means that both large a and small l can be obtained by choosing a long focal length of L6 or a short focal length of L3, thereby improving the imaging resolution. The measurement accuracy is influenced by not only the reference surface but also the collimated wavefront distortion. The incident wavefront has an inclination with reference and test mirrors such that the reference wavefront has a displacement with the test wavefront. Then, the collimated wavefront distortion contained in the reference and test wavefronts cannot cancel each other out. Consider a pair of coherent wavefronts as an example. The phase difference between the reference
where is the wavelength, d is the offset from the point source to the front focus of L3, and f3 , f4 , and f5 are the focal lengths of L3, L4, and L5, respectively. Eq. (2) indicates that the phase shifting can be achieved by varying d . If the offsets of the point-source array shown in Fig. 1 are different, then simultaneous phase shifting can be obtained. In particular, when the offset from the center of point-source array to the focus along x and y directions meet
x=
(4)
2.3. Imaging resolution and measurement accuracy
d 2f4 2 f32 f5 2
,
Because L4 and L5 comprise the beam expander, f4 is short. We have f4 . Thus, Eq. (5) can be satisfied easily by giving f52 f42 and f6 f6 + f4 ls > 0 .
where D is the cavity length between the reference and test mirrors and is the inclination of the beam incident on the reference mirror, which is determined by the beam expander, which is composed of L4 and L5 f4 / f5 . Becasue f3 d , = d/ f3 . Therefore, the phase such that = shift can be written according to Eq. (1) as
=
f52 f62 f7
where ls is the distance between L4 and L6, and the distance between L5 and the reference mirror is omitted; f6 , f7 , and f8 are the focal lengths of L6, L7, and L8, respectively, where L7 is placed at the rear focal plane of L6 and the front focal plane of L8. The condition that lD > f8 f82 / f7 should be satisfied to ensure that the four phase-shifting interferograms are non-overlapped according to the imaging analysis of Ref. [13]. Hence, based on Eq. (4)
2.2. Principle of simultaneous phase shifting
= 2D cos
f8 {f42 f5 f7 f8 Df42 f7 f8 + f52 [f62 (f7 f8 ) + f6 f7 f8 + f7 f8 (f4 ls )]}
, (3) 136
Optics and Laser Technology 111 (2019) 134–139
W. Zhu et al.
and test wavefronts can be expressed as
(p ) = =
T
T
(p ) +
(p ) +
+
2
+ 8
accuracy may be further improved by replacing it with a better reference mirror.
W (p )
D W (p ) s
,
4. Discussion
(6)
4.1. Design of the proposed interferometer
where p = (x , y ) is a spatial variable with Cartesian components x and y , T (p ) is the phase under test, W (p ) is the difference wavefront of the collimated wavefront, is the diameter of L5, and W (p )/ s is the partial derivative along the displacement direction. Eq. (6) indicates that the influence of the collimated wavefront distortion can be suppressed by increasing or decreasing . In the previous configuration [13], these conditions can be satisfied in a large aperture interf4 /f5 is referometer. However, in the proposed configuration, = duced with a beam expander such that a normal aperture interferometer can be applied. In addition, from Eq. (5),
8
f5 +
f52 (f6 + f4 f 42
If the beam expander is removed, the proposed interferometer is still a simultaneous phase-shifting Fizeau interferometer, but with a small aperture. Therefore, the design of the small-aperture interferometer should be optimized. According to Eq. (6), the influence of the collimated wavefront distortion can be suppressed by decreasing . This condition can be realized by increasing the focal length of L3. In additon, it is easier to control the collimated wavefront distortion by selecting a collimator with a long focal length. However, the focal length should not be too long; otherwise, the optical system will be extremely large because the relationship between the aperture a of L7 and the point-source array interval l is such that a = lf6 / f3. With respect to the designation of the beam expander, the magnification determines the measurement aperture of the proposed interferometer, which indicates that an optical zoom can be realized by varying the magnification. In addition, the larger the magnification, the higher the measurement accuracy.
ls )
max
W (p ) s
<
e,
(7)
which guarantees a high measurement accuracy, where max[·] is defined as the maximum and e is the maximum tolerable phase error. 3. Experiment
4.2. Imaging resolution
The experimental set up of the proposed interferometer system is depicted in Fig. 1. A He-Ne gas laser with a 632.8-nm wavelength was coupled in a fiber used as a point source. In order to produce a pointsource array with a small interval, the focal lengths of L1 and L2 are 100 mm and 60 mm, respectively, and a grating with a 38-μm pitch was designed to constrain the interval to be 2 mm. Then, the wavefronts emitted from the point-source array are transmitted through a collimating lens with a focal length of 200 mm before being expanded by a beam expander, where the focal lengths of L4 and L5 are −111.7 mm and 435.7 mm, respectively, and the expanded aperture is 80 mm. To guarantee measurement accuracy, an optical flat with a surface quality better than /20 was used as the reference mirror. A step plate with a diameter of 75 mm was placed 150 mm behind the reference mirror as the test mirror. The reference and test wavefronts reflected by the reference and test mirrors are returned to the imaging system, where a focal length of L6 was designed to be 300 mm such that the aperture of L7 can be increased from 2 mm to 3 mm, thereby improving the imaging resolution. ls is 70 mm, which meets the requirement of Eq. (5). The focal lengths of L7 and L8 were designed to be −40 mm and 35 mm, respectively, such that four phase-shifting interferograms are clearly imaged in the detector, from which the captured interferogram with 1200 × 1200 pixels is shown in Fig. 4(a). To make each phase step be π/2, we adjust the offset such that x = 0.16 mm and y = 0.32 mm according to Eq. (3). The four interferograms shown in Fig. 4(a) were denoted as I1, I2 , I3, and I4 by the addition of the corresponding phase shifts (0, /2, , 3 /2 ). Fig. 4(b) is the phase extracted from Fig. 4(a) using the phase-shifting algorithm [15] and phase-unwrapping method [16]. The phase step can be seen clearly, and its average height (AvgHgt) is 132.3 nm. Fig. 4(d) shows the test result of the same step plate from a ZYGO GPI interferometer, which is a conventional temporal phase-shifting Fizeau interferometer, where AvgHgt = 132.06 nm. To further compare the measurement accuracy, the upper phase steps of both results from Fig. 4(b) and (d) were selected. Both surfaces are shown in Fig. 4(c) and (e), where the peakvalley (PV) values are 0.040λ and 0.042λ, respectively, and the root mean square (RMS) values are 0.005λ and 0.003λ, respectively. Compared with the ZYGO GPI interferometer, the proposed interferometer achieves differences of 0.24 nm (AvgHgt), 0.002λ (PV) and 0.002λ (RMS). As the surface quality of the reference mirror is close to /20 , and the measured result has reached this limitation, the measurement
To analyze the imaging resolution, we used Zemax to simulate the imaging system of the proposed interferometer. According to the experimental parameters, the simulated imaging system is shown in Fig. 5(a), which shows that the primary limitation of the imaging resolution is the aperture of L7. More specifically, Fig. 5(b) and (c) present the modulation transfer function (MTF) with 2-mm and 3-mm diameters of L7, respectively. The spatial frequency is linearly proportional to the diameter. 4.3. Cavity length According to Eq. (6), the longer the cavity length D , the greater will be the collimated wavefront distortion. However, if D is too short, the offset ( x , y ) will be very large based on Eq. (3), thereby increasing the collimated wavefront distortion. In order to find the most suitable D , the square of the maximum displacement of the point-source array is given as 2 d max = ( x + l/2)2 + ( y + l/2)2
=
5 2f 34 f 54
64l2D2f 44
+
l2 2
+
3 f32 f52 8Df 42
,
(8)
where (m , n) = (0, 0) . Eq. (8) has 5 2f34 f54 /64l2D2f 44 = l2/2 . Thus, we have
10 f32 f52
D=
8l 2f42
a
minimum
,
when
(9)
which is the best cavity length for the measurement. 4.4. Error analysis The error sources mainly include the intensity aberration of four phase-shifting interferograms, the tilt of the test mirror, the distortion of the imaging system, the pixel mismatches of four phase-shifting interferograms, all of which have been discussed in Ref. [13], and the aberration of the collimated wavefront, which will be analyzed here. According to Eq. (8), the error term shown in Eq. (6) can be rewritten as
We =
137
8
3f 2 f 2 D 5f 4 f 4 l 2D 2 + 3 5 2 + 3 2 54 2 2 8 f4 64l f 4
W (p ) s
,
(10)
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Fig. 4. Experimental results: (a) the interferogram captured by the detector; (b) the phase extracted from (a); (c) the upper phase step of (b); (d) the test result of the same step plate from a ZYGO GPI interferometer; (e) the upper phase step of (d).
which indicates that We is an increasing function of D . Therefore, if the design of the proposed interferometer is not ideal, resulting in a large aberration to the collimated wavefront, the best approach is to reduce the cavity length even though is increased.
introduced into the previous simultaneous phase-shifting Fizeau interferometer, which can not only increase the imaging resolution, but also improve the measurement accuracy such that the proposed interferometer can be applied to normal aperture measurements. In addition, optical zoom can also be achieved at different measurement magnifications by the beam expander. In comparison, the proposed interferometer has a performance that is identical to that of the temporal phase-shifting Fizeau interferometer.
5. Conclusion This study presents an advanced simultaneous phase-shifting Fizeau interferometer. In the proposed configuration, a beam expander is
Fig. 5. Imaging system simulation: (a) the simulated imaging system; (b) the MTF with 2-mm diameter of L7; (c) the MTF with 3-mm diameter of L7. 138
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Acknowledgments
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This project was supported by the National Natural Science Foundation of China (U1731115, 61505082), the Natural Science Foundation of Jiangsu Province (SBK2015041354, BK20160154), the Fundamental Research Funds for the Central Universities (30916014112-003), and the China Scholarship Council. References [1] D. Malacara, Optical Shop Testing, John Wiley & Sons, 2007. [2] D.M. Sykora, M.L. Holmes, Dynamic measurements using a Fizeau interferometer, in: SPIE. 8082, 80821R-1-12, 2011. [3] T. Shirai, T.H. Barnes, T.G. Haskell, Adaptive wave-front correction by means of alloptical feedback interferometry, Opt. Lett. 25 (2000) 773–775. [4] C. Hernandez-Gomez, J.L. Collier, S.J. Hawkes, C.N. Danson, C.B. Edwards, D.A. Pepler, I.N. Ross, T.B. Winstone, Wave-front control of a large-aperture laser system by use of a static phase corrector, Appl. Opt. 39 (2000) 1954–1961. [5] K. Freischlad, R. Eng, J.B. Hadaway, Interferometer for testing in vibration environments, SPIE 4777 (2002) 311–322. [6] L.L. Deck, Environmentally friendly interferometry, SPIE 5532 (2004) 159–169.
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