Abstract

For white-light spectral interferometry, the phase information is usually retrieved via the Fourier transform method and the temporal phase-shifting method. In comparison, the synchronous phase-shifting method can be used to synchronously acquire interferometric signals with good accuracy and reduced noise. Therefore, it has potential for online measurement and is suitable for application in precision industries and for ultrahigh-speed measurement. In this work, a white-light spectral interferometer for synchronous phase shifting based on polarization interference was built, and the two-step phase-shifting algorithm was used to retrieve phase information. A variety of spectral interferometric signals were simulated based on the mathematical model of the two-step phase-shifting algorithm to illustrate the effects of differences in intensity and envelope shape, random noise, and phase-shift error on measurement of the absolute distance. Measurements of the absolute distance were conducted, and they indicated that the system had high accuracy.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (2)

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Y. Ghim and H. Rhee, “Instantaneous thickness measurement of multilayer films by single-shot angle-resolved spectral reflectometry,” Opt. Lett. 44, 5418–5421 (2019).
[Crossref]

2018 (2)

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

J. S. Li, X. X. Lu, Q. N. Zhang, B. B. Li, J. D. Tian, and L. Y. Zhong, “Dual-channel simultaneous spatial and temporal polarization phase-shifting interferometry,” Opt. Express 26, 4392–4400 (2018).
[Crossref]

2017 (1)

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

2015 (1)

2014 (1)

2013 (3)

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

R. L. Guo, B. L. Yao, P. Gao, J. W. Min, J. Han, X. Yu, M. Lei, S. H. Yan, Y. L. Yang, D. Dan, and T. Ye, “Parallel on-axis phase-shifting holographic phase microscopy based on reflective point-diffraction interferometer with long-term stability,” Appl. Opt. 52, 3484–3489 (2013).
[Crossref]

2010 (1)

2009 (3)

2008 (2)

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

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[Crossref]

2006 (2)

N. R. Sivakumar, B. Tan, and K. Venkatakrishnan, “Measurement of surface profile in vibrating environment with instantaneous phase shifting interferometry,” Opt. Commun. 257, 217–224 (2006).
[Crossref]

S. K. Debnath, M. P. Kothiyal, J. Schmit, and P. Hariharan, “Spectrally resolved white-light phase-shifting interference microscopy for thickness-profile measurements of transparent thin film layers on patterned substrates,” Opt. Express 14, 4662–4667 (2006).
[Crossref]

2005 (1)

2004 (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

2000 (2)

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[Crossref]

A. Harasaki and J. C. Wyant, “Fringe modulation skewing effect in white-light vertical scanning interferometry,” Appl. Opt. 39, 2101–2106 (2000).
[Crossref]

1995 (2)

U. Schnell, E. Zimmermann, and R. Dändliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[Crossref]

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

1985 (1)

1984 (1)

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

1979 (1)

W. Cleveland, “Robust locally weighted regression and smoothing scatterplots,” J. Am. Stat. Assoc. 74, 829–836 (1979).
[Crossref]

1971 (1)

Bai, F. Z.

Bhushan, B.

Bian, Y.

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Brock, N.

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, 6861–6868 (2005).
[Crossref]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Chen, Y. F.

Y. F. Chen and Y. L. Du, “One-shot surface profile measurement using polarized phase-shifting,” Proc. SPIE 7511, 328–332 (2009).
[Crossref]

Chen, Z.

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

Chlebus, R.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[Crossref]

Ciprian, D.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[Crossref]

Cleveland, W.

W. Cleveland, “Robust locally weighted regression and smoothing scatterplots,” J. Am. Stat. Assoc. 74, 829–836 (1979).
[Crossref]

Dan, D.

Dändliker, R.

U. Schnell, E. Zimmermann, and R. Dändliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[Crossref]

Debnath, S. K.

Deck, L. L.

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

Du, Y. L.

Y. F. Chen and Y. L. Du, “One-shot surface profile measurement using polarized phase-shifting,” Proc. SPIE 7511, 328–332 (2009).
[Crossref]

Feng, X. B.

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Fu, X.

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Fymat, A. L.

Gao, F.

Gao, P.

Ghim, Y.

Groot, P. J. D.

P. J. D. Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photon. 7, 1–65 (2015).
[Crossref]

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

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

P. J. D. Groot, “Phase shifting interferometry,” in Optical Measurement of Surface Topography, R. Leach, ed. (Springer, 2011), Chap. 8, pp. 167–185.

Guo, R. L.

Guo, T.

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Han, J.

Harasaki, A.

Hariharan, P.

Hayes, J.

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, 6861–6868 (2005).
[Crossref]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[Crossref]

Hlubina, P.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[Crossref]

Hu, X. T.

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Jiang, X.

Jiang, X. Q.

Kimbrough, B.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Koliopoulos, C. L.

Kothiyal, M. P.

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[Crossref]

Leach, R.

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Lei, M.

Li, B. B.

Li, F.

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Li, J. S.

Li, M. H.

T. Guo, L. Yuan, Z. Chen, M. H. Li, X. Fu, and X. T. Hu, “Single point Linnik white-light spectral microscopic interferometer for surface measurement,” Surf. Topogr. Metrol. Prop. 6, 034008 (2018).
[Crossref]

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

Liu, C. L.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Liu, S. G.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Lu, X. X.

Lunácek, J.

P. Hlubina, J. Luňáček, D. Ciprian, and R. Chlebus, “Windowed Fourier transform applied in the wavelength domain to process the spectral interference signals,” Opt. Commun. 281, 2349–2354 (2008).
[Crossref]

Ma, H. L.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Millerd, J.

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, 6861–6868 (2005).
[Crossref]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Min, J. W.

Moore, R.

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

Muhamedsalih, H.

Ni, L. F.

T. Guo, M. H. Li, Y. Zhou, L. F. Ni, X. Fu, and X. T. Hu, “Wavelength correction for thin film measurement in a microscopic white light spectral interferometer,” Optik 145, 188–201 (2017).
[Crossref]

North-Morris, M.

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, 6861–6868 (2005).
[Crossref]

J. Millerd, N. Brock, J. Hayes, M. North-Morris, B. Kimbrough, and J. Wyant, “Pixelated phase-mask dynamic interferometers,” Proc. SPIE 5531, 304–314 (2004).
[Crossref]

Novak, M.

Ramasamy, S.

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Rao, C. H.

Rhee, H.

Rinehart, M. T.

Schmit, J.

Schnell, U.

U. Schnell, E. Zimmermann, and R. Dändliker, “Absolute distance measurement with synchronously sampled white-light channelled spectrum interferometry,” Pure Appl. Opt. 4, 643–651 (1995).
[Crossref]

Schwider, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[Crossref]

Senin, N.

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Shaked, N. T.

Sivakumar, N. R.

N. R. Sivakumar, B. Tan, and K. Venkatakrishnan, “Measurement of surface profile in vibrating environment with instantaneous phase shifting interferometry,” Opt. Commun. 257, 217–224 (2006).
[Crossref]

Smythe, R.

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

Su, R.

X. B. Feng, N. Senin, R. Su, S. Ramasamy, and R. Leach, “Optical measurement of surface topographies with transparent coatings,” Opt. Laser Eng. 121, 261–270 (2019).
[Crossref]

Tan, B.

N. R. Sivakumar, B. Tan, and K. Venkatakrishnan, “Measurement of surface profile in vibrating environment with instantaneous phase shifting interferometry,” Opt. Commun. 257, 217–224 (2006).
[Crossref]

Tan, H.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Tang, D. W.

Tao, T. J.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Tian, J. D.

Venkatakrishnan, K.

N. R. Sivakumar, B. Tan, and K. Venkatakrishnan, “Measurement of surface profile in vibrating environment with instantaneous phase shifting interferometry,” Opt. Commun. 257, 217–224 (2006).
[Crossref]

Wang, K. W.

Wang, S. M.

Y. Bian, T. Guo, F. Li, S. M. Wang, X. Fu, and X. T. Hu, “Large step structure measurement by using white light interferometry based on adaptive scanning,” Proc. SPIE 8759, 87594T (2013).
[Crossref]

Wang, X.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Wax, A.

Weng, J. D.

J. D. Weng, T. J. Tao, S. G. Liu, H. L. Ma, X. Wang, C. L. Liu, and H. Tan, “Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range,” Rev. Sci. Instrum. 84, 113103 (2013).
[Crossref]

Wyant, J.

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, 6861–6868 (2005).
[Crossref]

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Figures (16)

Fig. 1.
Fig. 1. Schematic diagram of the Linnik microscopic white-light spectral interferometer.
Fig. 2.
Fig. 2. (a) and (b) Simulated spectral interferometric signals ${I_1}$ and ${I_2}$; (c) processed spectral signals ${I_1^\prime}$ and ${I_2^\prime}$; and (d) unwrapped phase.
Fig. 3.
Fig. 3. Plot of the recorded spectral signals.
Fig. 4.
Fig. 4. Processed spectral signals and unwrapped phase of (a) ${I_{2}} = {3}{I_1}$ and (b) ${I_{2}} = {5}{I_1}$.
Fig. 5.
Fig. 5. Plots of the errors in the absolute distances with different differences in spectral intensity.
Fig. 6.
Fig. 6. (a) Processed spectral signals and (b) unwrapped phase with uniform and Gaussian envelope shapes; (c) processed spectral signals and (d) unwrapped phase for same Gaussian shape but different central wavelengths.
Fig. 7.
Fig. 7. Plots of the errors in the absolute distances with different differences in envelope shape for (a) uniform and Gaussian shapes, and (b) both Gaussian shapes but different central wavelengths.
Fig. 8.
Fig. 8. Processed spectral signals and unwrapped phase with and without the normalization process for (a) intensity difference (${I_{2}} = {5}{I_1}$) with uniform envelope shapes, and for (b) uniform and Gaussian envelope shapes.
Fig. 9.
Fig. 9. Plots of the errors in the absolute distances with and without the normalization process for (a) intensity difference (${I_{2}} = {5}{I_1}$) with uniform envelope shapes, and for (b) uniform and Gaussian envelope shapes.
Fig. 10.
Fig. 10. Plots of the wrapped phase without the filtering process. (a) Spectral signals on adding Gaussian noise with an SNR of 50 dB; (b) spectral signals on adding Gaussian noise with an SNR of 40 dB; (c) spectral signals on adding Gaussian noise with an SNR of 30 dB.
Fig. 11.
Fig. 11. Plots of the absolute distances with and without filtering process. (a) Spectral signals on adding Gaussian noise with an SNR of 50 dB; (b) spectral signals on adding Gaussian noise with an SNR of 40 dB; (c) spectral signals on adding Gaussian noise with an SNR of 30 dB.
Fig. 12.
Fig. 12. Unwrapped phase of the phase-shift error of (a) −10°, (b) $ - {5}^\circ $, (c) $ + {5}^\circ$, and (c) $ + {10}^\circ$.
Fig. 13.
Fig. 13. Plots of the errors of the absolute distances with a phase-shift error of (a) $ - {10}^\circ $, (b) $ - {5}^\circ $, (c) $ + {5}^\circ$, and (c) $ + {10}^\circ$.
Fig. 14.
Fig. 14. (a) Recorded spectral signals ${I_1}$ and ${I_2}$; (b) processed spectral signals ${I_1^\prime}$ and ${I_2^\prime}$; (c) wrapped phase; (d) unwrapped phase.
Fig. 15.
Fig. 15. Measurement results. (a) Absolute distance; (b) differences in absolute distance.
Fig. 16.
Fig. 16. Measurement results for differences in absolute distance using different methods.

Equations (12)

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I = I r + I m + 2 I r I m cos ( φ ) ,
φ = 4 π d k = ϕ + 2 m π ,
d = 1 4 π Δ φ Δ k = 1 4 π Δ ( ϕ + 2 m π ) Δ k = 1 4 π Δ ϕ Δ k .
I = I r + I m + 2 I r I m cos ( φ + n Δ δ ) ,
G = G n G n 1 G i G 1 ,
E 10 = [ a 1 e i φ 1 0 ] , E 20 = [ 0 a 2 e i φ 2 ] .
E λ 4 = G λ 4 ( π 4 ) ( E 10 + E 20 ) = 1 2 [ 1 i       1 + i 1 + i       1 i ] [ a 1 e i φ 1 a 2 e i φ 2 ] .
E = G P E λ 4 = [ cos 2 α 1 2 sin 2 α 1 2 sin 2 α sin 2 α ] 1 2 [ 1 i       1 + i 1 + i       1 i ] [ a 1 e i φ 1 a 2 e i φ 2 ] = 2 2 [ a 1 e i ( φ 1 + α π 4 ) + a 2 e i ( φ 2 α + π 4 ) ] [ cos α sin α ] .
I = 1 2 [ a 1 cos ( θ 1 ) + a 2 cos ( θ 2 ) + i ( a 1 sin ( θ 1 ) + a 2 sin ( θ 2 ) ) ] 2 = 1 2 [ a 1 2 + a 2 2 + 2 a 1 a 2 cos ( φ 1 φ 2 2 α ) ] . θ 1 = φ 1 + α π 4 , θ 2 = φ 2 α + π 4 .
{ I 1 = I r + I m + 2 I r I m cos ( φ ) I 2 = I r + I m + 2 I r I m cos ( φ + π 2 ) ,
{ I 1 = 2 I r I m cos ( φ ) I 2 = 2 I r I m cos ( φ + π 2 ) .
φ = tan 1 ( I 2 I 1 ) .