Abstract

A general method is presented for spherical surface testing with unknown phase shifts based on a physical model of the interferometer cavity, which describes the phase shifts taking into account the rigid cavity motions and the radial imaging distortion of the interferometer. The captured interferograms are processed frame by frame with the regularized frequency-stabilizing method, so as to get the phase shifts between the frames. These phase shift data are subsequently fitted, and the initial estimations for the wavefront, direct current and interference contrast terms are calculated by the least-squares method. Specially, a simple way is proposed to find reasonable initial guess for numerical aperture (NA) of the test beam (when NA is unknown), so as to ensure the effectiveness of the above phase shift fitting procedure. Then, the wavefront result is further refined in an iterative way, by fitting the sequence of interferograms to the physical model of the interferometer cavity with the linear regression technique. Finally, the wavefront result related to the actual surface profile is retrieved after removing the aberrations due to the surface misalignment and the imaging distortion. Both simulations and experiments with the ZYGO interferometer have been carried out to validate the proposed method, with experimental accuracies better than 0.004λ RMS achieved. The proposed method provides a feasible way to spherical surface testing without the use of any phase-shifting devices, while retaining good accuracy and robust convergence performance.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  13. J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express 19(2), 584–596 (2011).
    [Crossref] [PubMed]
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    [Crossref]
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    [PubMed]
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    [Crossref] [PubMed]
  25. Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2014 (2)

2013 (3)

2012 (2)

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

2011 (4)

M. Liu, W. Yang, and W. Xu, “Calibration of measuring error caused by interferometric imaging distortion,” Opt. Precis. Eng. 19(10), 2349–2354 (2011).
[Crossref]

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express 19(2), 584–596 (2011).
[Crossref] [PubMed]

D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Misalignment aberrations calibration in testing of high-numerical-aperture spherical surfaces,” Appl. Opt. 50(14), 2024–2031 (2011).
[PubMed]

2010 (1)

2009 (2)

L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009).
[Crossref] [PubMed]

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

2008 (1)

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

2007 (1)

2005 (1)

2004 (1)

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

2003 (1)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

1997 (1)

1995 (2)

1994 (1)

1991 (1)

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1991).
[Crossref]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

1982 (1)

1980 (1)

1972 (1)

Álvarez-Herrero, A.

Belenguer, T.

Brock, N.

Chai, L.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

Chang, H. S.

Chen, C.

Chen, L.

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

Chen, Y. C.

Chu, G.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Creath, K.

Cuevas, F. J.

de Groot, P. J.

Deck, L. L.

Fan, B.

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

Fox, D. G.

Gao, Z.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

Groot, P.

Groot, P. J.

Gu, H.

Hariharan, P.

Hayes, J.

He, J.

Hock Soon, S.

Hou, X.

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

Ina, H.

Ji, F.

Jin, G.

Kemao, Q.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
[Crossref] [PubMed]

King, C. M.

Kobayashi, S.

Küchel, M. F.

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

Lee, C. C.

Li, K.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

Liang, C. W.

Lin, P. C.

Liu, D.

Liu, M.

M. Liu, W. Yang, and W. Xu, “Calibration of measuring error caused by interferometric imaging distortion,” Opt. Precis. Eng. 19(10), 2349–2354 (2011).
[Crossref]

Liu, Q.

Luo, Y.

Marroquin, J. L.

Millerd, J.

Moore, R. C.

North-Morris, M.

Novak, M.

Quiroga, J. A.

J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express 19(2), 584–596 (2011).
[Crossref] [PubMed]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Rimmer, M. P.

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1991).
[Crossref]

Servin, M.

Slaymaker, F. H.

Takeda, M.

Tan, Q.

Tian, C.

Vargas, J.

Wang, D.

Wang, H.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

Wang, Y.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Wu, F.

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

Wyant, J.

Xu, J.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

Xu, Q.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

Xu, W.

M. Liu, W. Yang, and W. Xu, “Calibration of measuring error caused by interferometric imaging distortion,” Opt. Precis. Eng. 19(10), 2349–2354 (2011).
[Crossref]

Yan, F.

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

Yang, W.

M. Liu, W. Yang, and W. Xu, “Calibration of measuring error caused by interferometric imaging distortion,” Opt. Precis. Eng. 19(10), 2349–2354 (2011).
[Crossref]

Yang, Y.

Yuan, Q.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Zeng, F.

Zhang, C.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Zhang, S.

Zhou, S.

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

Zhou, Y.

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

Zhu, R.

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

Zhuo, Y.

Appl. Opt. (10)

M. P. Rimmer, C. M. King, and D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt. 11(12), 2790–2796 (1972).
[Crossref] [PubMed]

R. C. Moore and F. H. Slaymaker, “Direct measurement of phase in a spherical-wave Fizeau interferometer,” Appl. Opt. 19(13), 2196–2200 (1980).
[Crossref] [PubMed]

K. Creath and P. Hariharan, “Phase-shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33(1), 24–25 (1994).
[Crossref] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
[Crossref] [PubMed]

P. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34(16), 2856–2863 (1995).
[Crossref] [PubMed]

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44(32), 6861–6868 (2005).
[Crossref] [PubMed]

L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. 48(20), 3948–3960 (2009).
[Crossref] [PubMed]

D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Misalignment aberrations calibration in testing of high-numerical-aperture spherical surfaces,” Appl. Opt. 50(14), 2024–2031 (2011).
[PubMed]

P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
[Crossref] [PubMed]

L. L. Deck, “Model-based phase shifting interferometry,” Appl. Opt. 53(21), 4628–4636 (2014).
[Crossref] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

Z. Gao, L. Chen, S. Zhou, and R. Zhu, “Computer-aided alignment for a reference transmission sphere of an interferometer,” Opt. Eng. 43(1), 69–74 (2004).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (2)

Q. Yuan, Z. Gao, Y. Zhou, G. Chu, and C. Zhang, “Calibration of phase-step nonuniformity in sub-nanometer-accuracy testing of high-numerical-aperture spherical surfaces,” Opt. Lasers Eng. 50(11), 1568–1574 (2012).
[Crossref]

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

Opt. Lett. (2)

Opt. Precis. Eng. (1)

M. Liu, W. Yang, and W. Xu, “Calibration of measuring error caused by interferometric imaging distortion,” Opt. Precis. Eng. 19(10), 2349–2354 (2011).
[Crossref]

Proc. SPIE (3)

F. Yan, B. Fan, X. Hou, and F. Wu, “A novel method for calibrating the image distortion of the interferometer,” Proc. SPIE 8145, 814516 (2012).

M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1991).
[Crossref]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Other (5)

D. Malacara, Optical Shop Testing, 3rd Edition (Wiley, 2007), Chap. 13, pp. 498–546.

K. M. Qian, Windowed Fringe Pattern Analysis (SPIE, 2013).

T. Dresel, “In situ determination of pixel mapping in interferometry,” ZYGO Corporation, US Patent 7, 495, 773 (2009).

M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th Edition (Cambridge University, 1999), Chap. 5, pp. 228–261.

D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for optical testing, 2nd Edition (CRC, 2005).

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Figures (8)

Fig. 1
Fig. 1

Block diagram of the proposed method

Fig. 2
Fig. 2

The simulated interferograms and surface profile. (a-c) The first three frames of simulated interferograms. (d) The simulated surface profile. The data shown in (d) are in radians.

Fig. 3
Fig. 3

The retrieved results by the proposed method. (a) The initial guess for NA. (b) The initial estimation of the wavefront data related to time t1. (c) The related residual errors in the wavefront data shown in (b). (d) The retrieved surface profile when imaging distortion is neglected. (e) A plot of the PV and RMS errors in retrieved surface profile versus iteration number by the proposed method. (f) The residual errors in the retrieved surface profile by the proposed method. The data shown in (b-c) and (d, f) are in radians.

Fig. 4
Fig. 4

The retrieved results by the method proposed in [1]. (a-c) The initial estimations of the direct current and interference contrast terms, as well as the wavefront data. (d) The erroneously retrieved surface profile in wrapped form. The data shown in (c-d) are in radians.

Fig. 5
Fig. 5

The retrieved results. (a-b) The retrieved surface profile and the corresponding error map, by the method proposed in [1]. (c) The resultant error map after removement of the numerical misalignment-style aberrations. All the data shown are in radians.

Fig. 6
Fig. 6

Comparisons of the stability performance between Deck’s method and our proposed method. (a) The successful convergence rate of Deck’s method with different amplitudes of random unknown phase shifts along z-axis. (b) The successful convergence rate of our proposed method.

Fig. 7
Fig. 7

(a) The first frame of the captured interferograms with unknown phase shifts. (b-c) The retrieved surface profile by the proposed method, with (b) or without (c) considering the imaging distortion. (d) The first frame of interferograms with well controlled phase shifts. (e) The retrieved surface profile based on the results given by ZYGO’s MetroPro software. (f) The difference between (b) and (e). The data shown in (b-c) and (e-f) are in radians.

Fig. 8
Fig. 8

The retrieved results by the proposed method in another two independent measurements. (a, d) show the first frame of the captured interferograms including unknown phase shifts in the two independent measurements, respectively. (b, e) show the retrieved surface profiles by the proposed method. (c, f) show the residual differences between these retrieved results and the retrieval result shown in Fig. 7(e). The data shown in (b-c) and (e-f) are in radians.

Tables (2)

Tables Icon

Table 1 The simulated time-dependent variables [kx(ti), ky (ti), kz(ti)], i = 1,2,…13, and the retrieved ones by the proposed method (units in radians)

Tables Icon

Table 2 The retrieved time-dependent variables [ k x ( t i ), k y ( t i ), k z ( t i ) ] , i=1,2,...,12 by the proposed method (units in radians)

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

I( x,y, t )=A( x,y )+V( x,y ) k=1 K (g) k1 cos[ kΘ( x,y, t ) ]
Θ( x,y, t )=Φ( x,y )+Δ( x,y, t )
Δ( x,y, t )=α( t )x+β( t )y+Ψ( t ) 1( x 2 + y 2 ) c 2
I( ρ,θ, t )=A( ρ,θ )+V( ρ,θ ) k=1 K (g) k1 cos[ kΘ( ρ,θ, t ) ]
Δ( ρ,θ, t )= k x ( t )ρcosθ+ k y ( t )ρsinθ+ k z ( t ) 1 ρ 2 N A 2
Δ( ρ,θ, t )= k x ( t )ρ( 1+ε ρ 2 )cosθ+ k y ( t )ρ( 1+ε ρ 2 )sinθ+ k z ( t ) 1 ρ 2 ( 1+ε ρ 2 ) 2 N A 2
Θ( x,y, t )=Θ( x,y, t 1 )+δ( x,y,t )
Θ( ρ,θ, t )=Θ( ρ,θ, t 1 )+δ( ρ,θ,t )
S1:{ δ ^ ( t i )= Θ ^ ( t i ) Θ ^ ( t 1 ) δ ^ ( t j )= Θ ^ ( t j ) Θ ^ ( t 1 ) , S2:{ δ ^ ( t i )= Θ ^ ( t i ) Θ ^ ( t 1 ) δ ^ ( t j )= Θ ^ ( t j ) Θ ^ ( t 1 ) , S3:{ δ ^ ( t i )= Θ ^ ( t i ) Θ ^ ( t 1 ) δ ^ ( t j )= Θ ^ ( t j ) Θ ^ ( t 1 ) , S4:{ δ ^ ( t i )= Θ ^ ( t i ) Θ ^ ( t 1 ) δ ^ ( t j )= Θ ^ ( t j ) Θ ^ ( t 1 )
I( ρ,θ,t )A( ρ,θ )+V( ρ,θ )cos[ Θ( ρ,θ, t ) ] =A( ρ,θ )+C( ρ,θ )cos[ δ ^ ( ρ,θ, t ) ]+S( ρ,θ )cos[ δ ^ ( ρ,θ, t ) ]
V( ρ,θ )= C ( ρ,θ ) 2 +S ( ρ,θ ) 2 , Θ w ( ρ,θ, t 1 )= tan 1 ( S( ρ,θ ) / C( ρ,θ ) )
δ( ρ,θ, t )=Δ k x ( t )ρ( 1+ε ρ 2 )cosθ+Δ k y ( t )ρ( 1+ε ρ 2 )sinθ +Δ k z ( t ) 1 ρ 2 ( 1+ε ρ 2 ) 2 N A 2
δ ^ fit ( ρ,θ, t )= p ^ ( t )+Δ k ^ x ( t )ρcosθ+Δ k ^ y ( t )ρsinθ+Δ k ^ z ( t ) 1 ρ 2 NA ^ 2
C( N A samp )= t | p ^ mod ( N A samp ,t ) | 2
p ^ mod ( N A samp ,t )=min{ | mod[ p ^ ( N A samp ,t ),2π ] |, | mod[ p ^ ( N A samp ,t ),2π ]2π | }
NA ^ =argmin[ C( N A samp ) ]
I ^ ( ρ,θ, t )= A ^ ( ρ,θ )+ V ^ ( ρ,θ ) k=1 K ( g ^ ) K1 cos[ k Θ ^ ( ρ,θ, t ) ]
Δ k ^ x,n ( t )=Δ k ^ x ( t )+ k x '( t ), Δ k ^ y,n ( t )=Δ k ^ y ( t )+ k y '( t ), Δ k ^ z,n ( t )=Δ k ^ z ( t )+ k z '( t ) ε ^ n = ε ^ +ε', NA ^ n = NA ^ +NA'
I ^ n ( ρ,θ, t )= I ^ ( ρ,θ, t )+[ ε' γ ε ( ρ,θ )+NA' γ NA ( ρ,θ ) ]H( ρ,θ,t )+ +[ k x '( t ) γ x ( ρ,θ )+ k y '( t ) γ y ( ρ,θ )+ k z '( t ) γ z ( ρ,θ ) ]H( ρ,θ,t )
γ x ( ρ,θ )=ρ( 1+ ε ^ ρ 2 )cosθ, γ y ( ρ,θ )=ρ( 1+ ε ^ ρ 2 )sinθ, γ z ( ρ,θ )= 1 ρ 2 ( 1+ ε ^ ρ 2 ) 2 NA ^ 2 , γ ε ( ρ,θ )=Δ k ^ x ( t ) ρ 3 cosθ+Δ k ^ y ( t ) ρ 3 sinθΔ k ^ z ( t ) ρ 4 ( 1+ ε ^ ρ 2 ) NA ^ 2 / γ z ( ρ,θ ) , γ NA = Δ k ^ z ( t ) ρ 2 ( 1+ ε ^ ρ 2 ) 2 NA ^ / γ z ( ρ,θ ) , H( ρ,θ,t )= V ^ ( ρ,θ ) k=1 K ( g ^ ) K1 sin[ k Θ ^ ( ρ,θ, t ) ]
χ( ρ,θ, t )= ρ,θ [ I( ρ,θ, t ) I ^ ( ρ,θ, t ) ] 2
Δ( ρ,θ, t ) a 0 Z 0 + a 1 Z 1 + a 2 Z 2 + a 3 Z 3 + a 10 Z 10 + a 21 Z 21 + a 36 Z 36
Δ( ρ,θ, t ) a 0 '+ a 1 ' Z 1 + a 2 ' Z 2 + a 3 ' Z 3 + a 6 ' Z 6 + a 7 ' Z 7 + a 10 ' Z 10 + a 21 ' Z 21 + a 36 ' Z 36
{ Z 0 =1 Z 1 =ρcosθ Z 2 =ρsinθ Z 3 =2 ρ 2 1 Z 6 =( 3 ρ 3 2ρ )sinθ , { Z 7 =( 3 ρ 3 2ρ )cosθ Z 10 =6 ρ 4 6 ρ 2 +1 Z 21 =20 ρ 6 30 ρ 4 +12 ρ 2 1 Z 36 =70 ρ 8 140 ρ 6 +90 ρ 4 20 ρ 2 +1
{ a 0 '= k z ( t )×[ 1( 1/ 4+ε/3 )N A 2 ( 1/ 24+ε/8 )N A 4 ( 1/ 64+ 3ε / 40 )N A 6 N A 8 / 128 ] a 1 '= k x ( t )×( 1+ 2ε /3 ) a 2 '= k y ( t )×( 1+ 2ε /3 ) a 3 '= k z ( t )×[ ( 1/ 4+ε/2 )N A 2 +( 1/ 16+ 9ε / 40 )N A 4 +( 9/ 320+ 3ε / 20 )N A 6 + N A 8 / 64 ] a 6 '= k y ( t )×ε /3 a 7 '= k x ( t )×ε /3 a 10 '= k z ( t )×[ εN A 2 / 6+( 1/ 48+ε/8 )N A 4 +( 1/ 64+ 3ε / 28 )N A 6 + 5N A 8 / 448 ] a 21 '= k z ( t )×[ εN A 4 / 40+( 1/ 320+ 3ε / 80 )N A 6 + N A 8 / 256 ] a 36 '= k z ( t )×( 3εN A 6 / 560+ N A 8 / 1792 )
k x '( t i )= k x ( t i )+Δ k x , k y '( t i )= k y ( t i )+Δ k y , k z '( t i )= k z ( t i )+Δ k z Φ ^ '( x,y )= Φ ^ ( x,y )+[ Δ k x x+Δ k y y+Δ k z 1( x 2 + y 2 )N A 2 ]
{ k x ( t i )=12π, k y ( t i )=0 k z ( t i )=π/ 4×( i7 ) + r z ×[ rand(1)0.5 ]

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