Abstract

A general method for reducing the influence of vibrations in phase-shifting interferometry corrects the surface phase map through a spectral analysis of a “phase-error pattern,” a plot of the interference intensity versus the measured phase, for each phase-shifted image. The method is computationally fast, applicable to any phase-shifting algorithm and interferometer geometry, has few restrictions on surface shape, and unlike spatial Fourier methods, high density spatial carrier fringes are not required, although at least a fringe of phase departure is recommended. Over a 100× reduction in vibrationally induced surface distortion is achieved for small amplitude vibrations on real data.

© 2009 Optical Society of America

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References

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  1. H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), Chap. 14.
    [CrossRef]
  2. P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354-365 (1995).
    [CrossRef]
  3. R. Doloca and R. Tutsch, “Vibration induced phase-shift interferometer,” Proc. SPIE 6292, 62920Y (2006).
  4. H. Martin, K. Wang, and X. Jiang, “Vibration compensating beam scanning interferometer for surface measurement,” Appl. Opt. 47, 888-893 (2008).
    [CrossRef] [PubMed]
  5. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).
  6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
    [CrossRef]
  7. T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt. 47, 3784-3788 (2008).
    [CrossRef] [PubMed]
  8. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  9. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier Transform,” Appl. Opt. 25, 1653-1660 (1986).
    [CrossRef] [PubMed]
  10. M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999-8005 (2006).
    [CrossRef] [PubMed]
  11. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
  12. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phaseshifting interferometers,” Appl. Opt. 39, 3894-3898 (2000).
    [CrossRef]
  13. G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321-7325 (1994).
    [CrossRef] [PubMed]
  14. P. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271-3279 (1990).
    [CrossRef] [PubMed]
  15. L. Deck, “Vibration-resistant phase-shifting interferometry,” Appl. Opt. 35, 6655-6662 (1996).
    [CrossRef] [PubMed]
  16. L. Deck and P. de Groot, “Punctuated quadrature phase-shifting interferometry,” Opt. Lett. 23, 19-21 (1998).
    [CrossRef]
  17. J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).
  18. X. Chen, M. Gramaglia, and J. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39, 585-591 (2000).
    [CrossRef]
  19. C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953-958 (1992).
    [CrossRef]
  20. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822-827 (1991).
    [CrossRef]
  21. K. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886-2894 (2001).
    [CrossRef]
  22. J. M. Huntley, “Suppression of phase errors from vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 15, 2233-2241 (1998).
    [CrossRef]
  23. The techniques described in this paper are protected by U.S. and foreign patents or patents pending.
  24. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723-4730 (1995).
    [CrossRef]
  25. P. de Groot and L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 70630K (2008).
    [CrossRef]
  26. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]
  27. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple errorcompensating phase calculation algorithm,” Appl. Opt. 26, 2504-2506 (1987).
    [CrossRef] [PubMed]
  28. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658-2663 (2000).
    [CrossRef]
  29. L. Deck, “Fourier-transform phase shifting interferometry,” Appl. Opt. 42, 2354-2365 (2003).
    [CrossRef] [PubMed]
  30. L. Deck, “Suppressing vibration errors in phase shifting interferometry,” Proc. SPIE 6704, 670402 (2007).
    [CrossRef]
  31. K. Creath, “Comparison of phase measuring algorithms,” Proc. SPIE 680, 19-28 (1986).
  32. For example, DynaFlect references manufactured by Zygo.
  33. L. Deck and P. de Groot, “High-speed non-contact profiler based on scanning white light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
    [CrossRef] [PubMed]
  34. exp[iucos(α)]=J0(u)+2∑k=1ikJk(u)cos().
  35. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542-551 (1990).
    [CrossRef]

2008

2007

L. Deck, “Suppressing vibration errors in phase shifting interferometry,” Proc. SPIE 6704, 670402 (2007).
[CrossRef]

2006

R. Doloca and R. Tutsch, “Vibration induced phase-shift interferometer,” Proc. SPIE 6292, 62920Y (2006).

M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999-8005 (2006).
[CrossRef] [PubMed]

2004

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

2003

2001

2000

1998

1996

1995

1994

1992

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

1991

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822-827 (1991).
[CrossRef]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

1990

1987

1986

1984

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

1983

1982

Bachor, H.-A.

Bokor, J.

Bone, D. J.

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Bruning, J.

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), Chap. 14.
[CrossRef]

Burow, R.

Chen, M.

Chen, X.

Creath, K.

K. Creath, “Comparison of phase measuring algorithms,” Proc. SPIE 680, 19-28 (1986).

de Groot, P.

Deck, L.

Doloca, R.

R. Doloca and R. Tutsch, “Vibration induced phase-shift interferometer,” Proc. SPIE 6292, 62920Y (2006).

Eiju, T.

Elssner, K.-E.

Farrell, C.

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Freischlad, K.

Goldberg, K.

Gramaglia, M.

Greivenkamp, J.

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

Grzanna, J.

Guo, H.

Han, G. S.

Hariharan, P.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Huntley, J. M.

Ina, H.

Jiang, X.

Kiire, T.

Kim, S. W.

Kitagawa, K.

Kobayashi, S.

Koliopoulos, C. L.

Lai, G.

Martin, H.

Merkel, K.

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Nakadate, S.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Ogawa, H.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Oreb, B. F.

Player, M.

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Sandeman, R. J.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Schreiber, H.

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), Chap. 14.
[CrossRef]

Schwider, J.

Shibuya, M.

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

Spolaczyk, R.

Sugiyama, M.

Suzuki, K.

Takeda, M.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Tutsch, R.

R. Doloca and R. Tutsch, “Vibration induced phase-shift interferometer,” Proc. SPIE 6292, 62920Y (2006).

Wang, K.

Wei, C.

Wizinowich, P.

Wyant, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Yatagai, T.

Yeazell, J.

Appl. Opt.

H. Martin, K. Wang, and X. Jiang, “Vibration compensating beam scanning interferometer for surface measurement,” Appl. Opt. 47, 888-893 (2008).
[CrossRef] [PubMed]

T. Kiire, S. Nakadate, and M. Shibuya, “Phase-shifting interferometer based on changing the direction of linear polarization orthogonally,” Appl. Opt. 47, 3784-3788 (2008).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier Transform,” Appl. Opt. 25, 1653-1660 (1986).
[CrossRef] [PubMed]

M. Sugiyama, H. Ogawa, K. Kitagawa, and K. Suzuki, “Single-shot surface profiling by local model fitting,” Appl. Opt. 45, 7999-8005 (2006).
[CrossRef] [PubMed]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phaseshifting interferometers,” Appl. Opt. 39, 3894-3898 (2000).
[CrossRef]

G. S. Han and S. W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33, 7321-7325 (1994).
[CrossRef] [PubMed]

P. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271-3279 (1990).
[CrossRef] [PubMed]

L. Deck, “Vibration-resistant phase-shifting interferometry,” Appl. Opt. 35, 6655-6662 (1996).
[CrossRef] [PubMed]

X. Chen, M. Gramaglia, and J. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39, 585-591 (2000).
[CrossRef]

K. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40, 2886-2894 (2001).
[CrossRef]

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723-4730 (1995).
[CrossRef]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421-3432 (1983).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple errorcompensating phase calculation algorithm,” Appl. Opt. 26, 2504-2506 (1987).
[CrossRef] [PubMed]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658-2663 (2000).
[CrossRef]

L. Deck, “Fourier-transform phase shifting interferometry,” Appl. Opt. 42, 2354-2365 (2003).
[CrossRef] [PubMed]

L. Deck and P. de Groot, “High-speed non-contact profiler based on scanning white light interferometry,” Appl. Opt. 33, 7334-7338 (1994).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase shifting interferometry,” Meas. Sci. Technol. 3, 953-958 (1992).
[CrossRef]

Opt. Commun.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Opt. Eng.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-364 (1984).

J. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350-352 (1984).

Opt. Lett.

Proc. SPIE

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. Wyant, “Pixellated phasemask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

R. Doloca and R. Tutsch, “Vibration induced phase-shift interferometer,” Proc. SPIE 6292, 62920Y (2006).

P. de Groot and L. Deck, “New algorithms and error analysis for sinusoidal phase shifting interferometry,” Proc. SPIE 7063, 70630K (2008).
[CrossRef]

L. Deck, “Suppressing vibration errors in phase shifting interferometry,” Proc. SPIE 6704, 670402 (2007).
[CrossRef]

Other

K. Creath, “Comparison of phase measuring algorithms,” Proc. SPIE 680, 19-28 (1986).

For example, DynaFlect references manufactured by Zygo.

The techniques described in this paper are protected by U.S. and foreign patents or patents pending.

exp[iucos(α)]=J0(u)+2∑k=1ikJk(u)cos().

H. Schreiber and J. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, 2007), Chap. 14.
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Typical PSI measurement system.

Fig. 2
Fig. 2

Phase-error patterns from a simulated surface with at least one fringe of departure for both vibration free and 0.5 rad amplitude vibrations.

Fig. 3
Fig. 3

Phase height profile of the simulated surface.

Fig. 4
Fig. 4

Interferogram intensity of the simulated surface (light gray) and measured PSI phase error due to vibration (dark line). The phase error shows the expected 2-cycle distortion characteristic.

Fig. 5
Fig. 5

Phase-error pattern generated using the intensities of the first frame and the measured phase. Each data point corresponds to an individual pixel.

Fig. 6
Fig. 6

Power spectrum of the phase-error pattern shown in Fig. 5 showing the predicted dependence on odd-order harmonics.

Fig. 7
Fig. 7

Residual phase error after applying the VC correction to 3rd order (dark line near zero) compared with the original phase error (gray line).

Fig. 8
Fig. 8

Residual phase-error profiles for standard 5-frame PSI phase map and the VC phase map for two values of the harmonic order. The vibration amplitude varied linearly across the profile.

Fig. 9
Fig. 9

RMS vibration sensitivity for small amplitude vibrations compared to PSI algorithm sensitivity for the 5 frame (left) and 13 frame (right) PSI algorithms. The vibration frequency is normalized to the sample rate. The vibration amplitude was 0.1 rad .

Fig. 10
Fig. 10

RMS vibration sensitivity for large amplitude vibrations compared to PSI algorithm sensitivity for the 5 frame (left) and 13 frame (right) PSI algorithms. The vibration frequency is normalized to the sample rate. The vibration amplitude was 1 rad .

Fig. 11
Fig. 11

13-frame PSI measurement of a vibrated flat cavity with about 5 fringes of tilt containing ripple with 25 nm amplitude before (left) and after (right) the VC method to 3rd order is applied.

Fig. 12
Fig. 12

13-frame PSI measurement of a vibrated spherical cavity with 4 fringes of departure containing ripple with 25 nm amplitude before (left) and after (right) the VC method to 3rd order is applied.

Fig. 13
Fig. 13

Surface profile of a poorly mounted flat that incurred significant tilt during the acquisition. The left figure is the surface using 13-frame PSI, the center is with 3rd order VC without spatial dependence, and the right figure is 3rd order VC incorporating spatial dependence.

Fig. 14
Fig. 14

VC applied to data from a phase-measuring microscope. The fringe pattern is shown at left. The center image shows the measured surface profile without VC and the right image is with VC.

Tables (1)

Tables Icon

Table 1 Sampling Coefficients for the Two PSI Algorithms Used in This Paper

Equations (32)

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exp [ i Φ ( x ) ] = k = 0 g k exp [ i ( 2 k + 1 ) Φ ^ ( x ) ] k = 1 g ¯ k exp [ i ( 2 k 1 ) Φ ^ ( x ) ] 2 k 1 ,
c ( Φ ^ , t ) = I 0 V 2 { exp [ i β ( t ) ] [ k = 0 g k exp [ i ( 2 k + 1 ) Φ ^ ] k = 1 g ¯ k exp [ i ( 2 k 1 ) Φ ^ ] 2 k 1 ] + exp [ i β ( t ) ] [ k = 0 g ¯ k exp [ i ( 2 k + 1 ) Φ ^ ] k = 1 g k exp [ i ( 2 k 1 ) Φ ^ ] 2 k 1 ] } .
C ( K , t ) = c ( Φ ^ , t ) exp ( i K Φ ^ ) d Φ ^ ,
C ( K , t ) = I 0 V 2 { exp [ i β ( t ) ] [ k = 0 g k δ ( K 2 k 1 ) + k = 1 g ¯ k δ ( K + 2 k 1 ) 2 k 1 ] + exp [ i β ( t ) ] [ k = 0 g ¯ k δ ( K + 2 k + 1 ) + k = 1 g k δ ( K 2 k + 1 ) 2 k 1 ] } .
C ( 2 k + 1 , t ) = g k exp [ i β ( t ) ] g k + 1 2 k + 1 exp [ i β ( t ) ] .
Im [ k = 0 κ C ( 2 k + 1 , t ) exp [ i ( 2 k + 1 ) β ( t ) ] ( 1 2 ) ! ( k 1 2 ) ! 2 k ] = 0.
exp ( i Δ PSI ) = exp ( i ( Φ ^ Φ ) ) = η + μ ¯ exp ( 2 i Φ ) | η exp ( i Φ ) + μ ¯ exp ( i Φ ) | ,
exp ( i Δ VC ) = g 0 k = 1 ( g k exp ( i 2 k Φ ^ ) g ¯ k exp ( i 2 k Φ ^ ) 2 k 1 ) ,
I ( x , t ) = I 0 [ 1 + p cos ( ω p t + γ ) ] [ 1 + V cos ( Φ ( x ) + ω 0 t ) ] ,
c ( Φ ^ , t ) G ( t ) + k = 0 3 g k ( t ) exp ( i k Φ ^ ) + k = 1 3 g ¯ k ( t ) exp ( i k Φ ^ ) ,
C ( K , t ) = k = 0 3 g k ( t ) δ ( K k ) + k = 1 3 g ¯ k ( t ) δ ( K + k ) .
I ( x , t ) = I 0 { 1 + V cos [ Φ ( x ) + ω 0 t + r cos ( ω v t + α ) ] } ,
s ( x , t ) = I 0 V 2 { exp [ i Φ ( x ) ] exp [ i β ( t ) ] + exp [ i Φ ( x ) ] exp [ i β ( t ) ] } ,
1 τ t τ 2 t + τ 2 e i g t d t = e i g t sinc ( g τ / 2 ) ,
s ( x , t ) = I 0 V 2 J 0 ( r ) sinc ( ω 0 τ 2 ) { exp [ i Φ ( x ) + i ω 0 t ] + exp [ i Φ ( x ) i ω 0 t ] } + I 0 V 2 k = 1 i k J k ( r ) sinc [ τ ( ω 0 + k ω v ) 2 ] exp [ i Φ ( x ) + i k α ] exp [ i ( ω 0 + k ω v ) t ] + I 0 V 2 k = 1 i k J k ( r ) sinc [ τ ( ω 0 k ω v ) 2 ] exp [ i Φ ( x ) i k α ] exp [ i ( ω 0 k ω v ) t ] I 0 V 2 k = 1 ( i ) k J k ( r ) sinc [ τ ( ω 0 + k ω v ) 2 ] exp [ i Φ ( x ) + i k α ] exp [ i ( ω 0 + k ω v ) t ] + I 0 V 2 k = 1 ( i ) k J k ( r ) sinc [ τ ( ω 0 k ω v ) 2 ] exp [ i Φ ( x ) i k α ] exp [ i ( ω 0 k ω v ) t ] .
S ( ω ) = s ( t ) exp ( i ω t ) d t ,
S ( x , ω ) = I 0 V 2 J 0 ( r ) sinc ( ω 0 τ 2 ) { exp [ i Φ ( x ) ] δ ( ω ω 0 ) + exp [ i Φ ( x ) ] δ ( ω + ω 0 ) } + I 0 V 2 k = 1 i k J k ( r ) sinc [ τ ( ω 0 + k ω v ) 2 ] exp [ i ( Φ ( x ) + k α ) ] δ ( ω ω 0 k ω v ) + I 0 V 2 k = 1 i k J k ( r ) sinc [ τ ( ω 0 k ω v ) 2 ] exp [ i ( Φ ( x ) k α ) ] δ ( ω ω 0 + k ω v ) + I 0 V 2 k = 1 ( i ) k J k ( r ) sinc [ τ ( ω 0 + k ω v ) 2 ] exp [ i ( Φ ( x ) + k α ) ] δ ( ω + ω 0 k ω v ) + I 0 V 2 k = 1 ( i ) k J k ( r ) sinc [ τ ( ω 0 + k ω v ) 2 ] exp [ i ( Φ ( x ) + k α ) ] δ ( ω + ω 0 k ω v ) ,
Φ ^ ( x ) = arg [ S ( x , ω 0 ) ] = atan { Im [ S ( x , ω 0 ) ] Re [ S ( x , ω 0 ) ] } .
S ( x , ω 0 ) = η exp [ i Φ ( x ) ] + μ ¯ exp [ i Φ ( x ) ] ,
η = I 0 V 2 J 0 ( r ) sinc ( ω 0 τ 2 ) δ ( 0 ) ,
μ ¯ = I 0 V 2 k = 1 ( i ) k J k ( r ) sinc ( τ ( k ω v ω 0 ) 2 ) exp ( i k α ) δ ( ω v 2 ω 0 / k ) .
exp ( i Φ ^ ( x ) ) = η exp [ i Φ ( x ) ] + μ ¯ exp [ i Φ ( x ) ] | η exp [ i Φ ( x ) ] + μ ¯ exp [ i Φ ( x ) ] | .
exp [ i Φ ( x ) ] = ± exp [ i Φ ^ ( x ) ] η ¯ | η | 1 z ¯ exp [ 2 i Φ ^ ( x ) ] 1 z exp [ 2 i Φ ^ ( x ) ] ,
F ( ω ) = j = 0 N 1 c j exp ( i ω j ) ,
η = I 0 V 2 J 0 ( r ) sinc ( ω 0 τ 2 ) F ( 0 ) ,
μ ¯ = I 0 V 2 k = 1 ( i ) k J k ( r ) sinc ( τ ( k ω v ω 0 ) 2 ) exp ( i k α ) F ( ω v 2 ω 0 / k ) .
exp [ i Φ ( x ) ] = exp [ i Φ ^ ( x ) ] n = 0 p n exp [ i 2 n Φ ^ ( x ) ] m = 0 q m exp [ i 2 m Φ ^ ( x ) ] ,
p n = 1 2 ! ( 1 2 n ) ! n ! ( z ¯ ) n and q m = 1 2 ! ( 1 2 m ) ! m ! ( z ) m .
exp [ i Φ ( x ) ] = k = g k exp [ i ( 2 k + 1 ) Φ ^ ( x ) ] ,
g k = { n = 0 p n q n + k k 0 n = 0 p n k q n k < 0 .
g k ( 2 k 1 ) g ¯ k ,
exp [ i Φ ( x ) ] = k = 0 g k exp [ i ( 2 k + 1 ) Φ ^ ( x ) ] k = 1 g ¯ k exp [ i ( 2 k 1 ) Φ ^ ( x ) ] 2 k 1 .

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