Abstract

The theory and experimental results of a computation time-saving mirror image suppression method in Fourier-domain optical coherence tomography, which utilizes the property of reversed system phase shift between the real and mirror images, for differentiating one from the other are demonstrated. By solving a set of two equations based on a reasonable approximation, the real image signal can be obtained. The theoretical backgrounds and the improved real image quality of the average and iteration procedures in this method are particularly illustrated. Also, the mirror image suppression ratios under various process conditions, including different process iteration numbers and different system phase shifts between two neighboring A-mode scans, are evaluated. Meanwhile, the mirror image suppression results based on our method are compared with those obtained from the widely used BM-scan technique. It is found that when a process procedure of two iterations is used, the mirror image suppression quality based on our method can be higher than that obtained from the BM-scan technique. The computation time of our method is significantly shorter than that of the BM-scan technique.

© 2012 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
    [CrossRef]
  4. Y. Yasuno, S. Makita, T. Endo, G. Aoki, H. Sumimura, M. Itoh, and T. Yatagai, “One-shot-phase-shifting Fourier domain optical coherence tomography by reference wavefront tilting,” Opt. Express 12(25), 6184–6191 (2004).
    [CrossRef] [PubMed]
  5. R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase shifting algorithm to achieve high speed long depth range probing by frequency domain optical coherence tomography,” Opt. Lett. 28(22), 2201–2203 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers,” Opt. Express 13(3), 957–967 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. S. Witte, M. Baclayon, E. J. Peterman, R. F. Toonen, H. D. Mansvelder, and M. L. Groot, “Single-shot two-dimensional full-range optical coherence tomography achieved by dispersion control,” Opt. Express 17(14), 11335–11349 (2009).
    [CrossRef] [PubMed]
  25. B. Hermann, B. Hofer, C. Meier, and W. Drexler, “Spectroscopic measurements with dispersion encoded full range frequency domain optical coherence tomography in single- and multilayered non-scattering phantoms,” Opt. Express 17(26), 24162–24174 (2009).
    [CrossRef] [PubMed]
  26. B. Hofer, B. Povazay, A. Unterhuber, L. Wang, B. Hermann, S. Rey, G. Matz, and W. Drexler, “Fast dispersion encoded full range optical coherence tomography for retinal imaging at 800 nm and 1060 nm,” Opt. Express 18(5), 4898–4919 (2010).
    [CrossRef] [PubMed]
  27. Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography,” Appl. Opt. 45(8), 1861–1865 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  30. S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008).
    [CrossRef] [PubMed]
  31. K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express 17(19), 16820–16833 (2009).
    [CrossRef] [PubMed]
  32. F. Jaillon, S. Makita, M. Yabusaki, and Y. Yasuno, “Parabolic BM-scan technique for full range Doppler spectral domain optical coherence tomography,” Opt. Express 18(2), 1358–1372 (2010).
    [CrossRef] [PubMed]
  33. B. Baumann, M. Pircher, E. Götzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15(20), 13375–13387 (2007).
    [CrossRef] [PubMed]
  34. L. An and R. K. Wang, “Use of a scanner to modulate spatial interferograms for in vivo full-range Fourier-domain optical coherence tomography,” Opt. Lett. 32(23), 3423–3425 (2007).
    [CrossRef] [PubMed]
  35. R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32(23), 3453–3455 (2007).
    [CrossRef] [PubMed]
  36. C. T. Wu, T. T. Chi, C. K. Lee, Y. W. Kiang, C. C. Yang, and C. P. Chiang, “Method for suppressing the mirror image in Fourier-domain optical coherence tomography,” Opt. Lett. 36(15), 2889–2891 (2011).
    [CrossRef] [PubMed]

2011 (2)

2010 (4)

2009 (5)

2008 (3)

2007 (6)

2006 (4)

2005 (4)

2004 (4)

2003 (2)

2002 (1)

1999 (1)

A. F. Fercher, R. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173–178 (1999).
[CrossRef]

An, L.

Aoki, G.

Applegate, B. E.

Bachmann, A.

Baclayon, M.

Bajraszewski, T.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase shifting algorithm to achieve high speed long depth range probing by frequency domain optical coherence tomography,” Opt. Lett. 28(22), 2201–2203 (2003).
[CrossRef] [PubMed]

Baumann, B.

Bonesi, M.

Bouma, B.

Chang, S.

Chen, M.

Chen, Z.

Cheng, H. C.

H. C. Cheng, J. F. Huang, and Y. H. Hsieh, “Numerical analysis of one-shot full-range FD-OCT system based on orthogonally polarized light,” Opt. Commun. 282(14), 3040–3045 (2009).
[CrossRef]

Chi, T. T.

Chiang, C. P.

Choma, M. A.

Dallas, W.

Davis, A. M.

A. M. Davis, M. A. Choma, and J. A. Izatt, “Heterodyne swept-source optical coherence tomography for complete complex conjugate ambiguity removal,” J. Biomed. Opt. 10(6), 064005 (2005).
[CrossRef] [PubMed]

de Boer, J.

Dhalla, A. H.

Ding, Z.

Drexler, W.

Dufour, M. L.

Endo, T.

Fabritius, T.

Fercher, A. F.

Flueraru, C.

Gorczynska, I.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

Götzinger, E.

Groot, M. L.

Hermann, B.

Hitzenberger, C.

Hitzenberger, C. K.

Hofer, B.

Hsieh, Y. H.

H. C. Cheng, J. F. Huang, and Y. H. Hsieh, “Numerical analysis of one-shot full-range FD-OCT system based on orthogonally polarized light,” Opt. Commun. 282(14), 3040–3045 (2009).
[CrossRef]

Hsu, K.

Huang, J. F.

H. C. Cheng, J. F. Huang, and Y. H. Hsieh, “Numerical analysis of one-shot full-range FD-OCT system based on orthogonally polarized light,” Opt. Commun. 282(14), 3040–3045 (2009).
[CrossRef]

Itoh, M.

Izatt, J. A.

Jaillon, F.

Jung, W.

Kane, D. J.

Kiang, Y. W.

Kowalczyk, A.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002).
[CrossRef] [PubMed]

Lamouche, G.

Lasser, T.

Lee, C. K.

Lee, K. S.

Leitgeb, R.

Leitgeb, R. A.

Makita, S.

Mansvelder, H. D.

Mao, Y.

Matz, G.

Meemon, P.

Meier, C.

Meng, J.

Michaely, R.

Nelson, J.

Nelson, J. S.

Peterman, E. J.

Peterson, K. A.

Pircher, M.

Povazay, B.

Rey, S.

Rolland, J. P.

Sarunic, M.

Sarunic, M. V.

Sattmann, H.

A. F. Fercher, R. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173–178 (1999).
[CrossRef]

Sekhar, S. C.

Sherif, S.

Sumimura, H.

Szkulmowski, M.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

Tao, Y. K.

Targowski, P.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

Tearney, G.

Toonen, R. F.

Torzicky, T.

Unterhuber, A.

Vakhtin, A. B.

Vergnole, S.

Wang, K.

Wang, L.

Wang, R. K.

Witte, S.

Wojtkowski, M.

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002).
[CrossRef] [PubMed]

A. F. Fercher, R. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173–178 (1999).
[CrossRef]

Wu, C. T.

Yabusaki, M.

Yang, C.

Yang, C. C.

Yasuno, Y.

Yatagai, T.

Yun, S.

Zeng, Y.

Zhang, J.

Zhao, M.

Zotter, S.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. K. Wang, “In vivo full range complex Fourier domain optical coherence tomography,” Appl. Phys. Lett. 90(5), 054103 (2007).
[CrossRef]

Biomed. Opt. Express (1)

J. Biomed. Opt. (1)

A. M. Davis, M. A. Choma, and J. A. Izatt, “Heterodyne swept-source optical coherence tomography for complete complex conjugate ambiguity removal,” J. Biomed. Opt. 10(6), 064005 (2005).
[CrossRef] [PubMed]

Opt. Commun. (2)

P. Targowski, M. Wojtkowski, A. Kowalczyk, T. Bajraszewski, M. Szkulmowski, and I. Gorczynska, “Complex spectral OCT in human eye imaging in vivo,” Opt. Commun. 229(1-6), 79–84 (2004).
[CrossRef]

H. C. Cheng, J. F. Huang, and Y. H. Hsieh, “Numerical analysis of one-shot full-range FD-OCT system based on orthogonally polarized light,” Opt. Commun. 282(14), 3040–3045 (2009).
[CrossRef]

Opt. Express (14)

E. Götzinger, M. Pircher, R. Leitgeb, and C. Hitzenberger, “High speed full range complex spectral domain optical coherence tomography,” Opt. Express 13(2), 583–594 (2005).
[CrossRef] [PubMed]

M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers,” Opt. Express 13(3), 957–967 (2005).
[CrossRef] [PubMed]

A. Bachmann, R. Leitgeb, and T. Lasser, “Heterodyne Fourier domain optical coherence tomography for full range probing with high axial resolution,” Opt. Express 14(4), 1487–1496 (2006).
[CrossRef] [PubMed]

S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express 12(20), 4822–4828 (2004).
[CrossRef] [PubMed]

J. Zhang, W. Jung, J. Nelson, and Z. Chen, “Full range polarization-sensitive Fourier domain optical coherence tomography,” Opt. Express 12(24), 6033–6039 (2004).
[CrossRef] [PubMed]

Y. Yasuno, S. Makita, T. Endo, G. Aoki, H. Sumimura, M. Itoh, and T. Yatagai, “One-shot-phase-shifting Fourier domain optical coherence tomography by reference wavefront tilting,” Opt. Express 12(25), 6184–6191 (2004).
[CrossRef] [PubMed]

B. Baumann, M. Pircher, E. Götzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15(20), 13375–13387 (2007).
[CrossRef] [PubMed]

S. Makita, T. Fabritius, and Y. Yasuno, “Full-range, high-speed, high-resolution 1 microm spectral-domain optical coherence tomography using BM-scan for volumetric imaging of the human posterior eye,” Opt. Express 16(12), 8406–8420 (2008).
[CrossRef] [PubMed]

B. Hofer, B. Povazay, B. Hermann, A. Unterhuber, G. Matz, and W. Drexler, “Dispersion encoded full range frequency domain optical coherence tomography,” Opt. Express 17(1), 7–24 (2009).
[CrossRef] [PubMed]

S. Witte, M. Baclayon, E. J. Peterman, R. F. Toonen, H. D. Mansvelder, and M. L. Groot, “Single-shot two-dimensional full-range optical coherence tomography achieved by dispersion control,” Opt. Express 17(14), 11335–11349 (2009).
[CrossRef] [PubMed]

K. Wang, Z. Ding, Y. Zeng, J. Meng, and M. Chen, “Sinusoidal B-M method based spectral domain optical coherence tomography for the elimination of complex-conjugate artifact,” Opt. Express 17(19), 16820–16833 (2009).
[CrossRef] [PubMed]

B. Hermann, B. Hofer, C. Meier, and W. Drexler, “Spectroscopic measurements with dispersion encoded full range frequency domain optical coherence tomography in single- and multilayered non-scattering phantoms,” Opt. Express 17(26), 24162–24174 (2009).
[CrossRef] [PubMed]

F. Jaillon, S. Makita, M. Yabusaki, and Y. Yasuno, “Parabolic BM-scan technique for full range Doppler spectral domain optical coherence tomography,” Opt. Express 18(2), 1358–1372 (2010).
[CrossRef] [PubMed]

B. Hofer, B. Povazay, A. Unterhuber, L. Wang, B. Hermann, S. Rey, G. Matz, and W. Drexler, “Fast dispersion encoded full range optical coherence tomography for retinal imaging at 800 nm and 1060 nm,” Opt. Express 18(5), 4898–4919 (2010).
[CrossRef] [PubMed]

Opt. Lett. (13)

K. S. Lee, P. Meemon, W. Dallas, K. Hsu, and J. P. Rolland, “Dual detection full range frequency domain optical coherence tomography,” Opt. Lett. 35(7), 1058–1060 (2010).
[CrossRef] [PubMed]

S. Zotter, M. Pircher, E. Götzinger, T. Torzicky, M. Bonesi, and C. K. Hitzenberger, “Sample motion-insensitive, full-range, complex, spectral-domain optical-coherence tomography,” Opt. Lett. 35(23), 3913–3915 (2010).
[CrossRef] [PubMed]

Y. K. Tao, M. Zhao, and J. A. Izatt, “High-speed complex conjugate resolved retinal spectral domain optical coherence tomography using sinusoidal phase modulation,” Opt. Lett. 32(20), 2918–2920 (2007).
[CrossRef] [PubMed]

L. An and R. K. Wang, “Use of a scanner to modulate spatial interferograms for in vivo full-range Fourier-domain optical coherence tomography,” Opt. Lett. 32(23), 3423–3425 (2007).
[CrossRef] [PubMed]

R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32(23), 3453–3455 (2007).
[CrossRef] [PubMed]

S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008).
[CrossRef] [PubMed]

C. T. Wu, T. T. Chi, C. K. Lee, Y. W. Kiang, C. C. Yang, and C. P. Chiang, “Method for suppressing the mirror image in Fourier-domain optical coherence tomography,” Opt. Lett. 36(15), 2889–2891 (2011).
[CrossRef] [PubMed]

J. Zhang, J. S. Nelson, and Z. Chen, “Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator,” Opt. Lett. 30(2), 147–149 (2005).
[CrossRef] [PubMed]

A. B. Vakhtin, K. A. Peterson, and D. J. Kane, “Resolving the complex conjugate ambiguity in Fourier-domain OCT by harmonic lock-in detection of the spectral interferogram,” Opt. Lett. 31(9), 1271–1273 (2006).
[CrossRef] [PubMed]

M. V. Sarunic, B. E. Applegate, and J. A. Izatt, “Real-time quadrature projection complex conjugate resolved Fourier domain optical coherence tomography,” Opt. Lett. 31(16), 2426–2428 (2006).
[CrossRef] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002).
[CrossRef] [PubMed]

M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3 x 3 fiber-optic couplers,” Opt. Lett. 28(22), 2162–2164 (2003).
[CrossRef] [PubMed]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase shifting algorithm to achieve high speed long depth range probing by frequency domain optical coherence tomography,” Opt. Lett. 28(22), 2201–2203 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

A. F. Fercher, R. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173–178 (1999).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Setup of the SD-OCT system for sample scanning.

Fig. 2
Fig. 2

OCT images of human skin on a finger, including (a) the un-processed image with the overlapping real and mirror images, (b) the image after one iteration of our mirror image suppression method, (c) the image after two iterations of our mirror image suppression method, and (d) the image processed with the BM-scan method. Both scan and process phase shifts are 90 degrees.

Fig. 3
Fig. 3

OCT images of hamster pouch mucosa, including (a) the un-processed image with the overlapping real and mirror images, (b) the image after one iteration of our mirror image suppression method, (c) the image after two iterations of our mirror image suppression method, and (d) the image processed with the BM-scan method. Both scan and process phase shifts are 90 degrees.

Fig. 4
Fig. 4

OCT scanning and processed results of the USAF 1951 test target (R3L3S1N). (a): The image of the test target with the OCT scanning trace indicated by the vertical (pink) line; (b): The OCT scanning image before the process of mirror image suppression with the labeled slit widths. (c)-(e): The OCT images after the processes of mirror image suppression with 1, 2, and 3 iterations, respectively.

Fig. 5
Fig. 5

Mirror image suppression ratios as functions of the scan phase shift when the process phase shift is always equal to the scan phase shift under different process conditions, including the cases of one (1), two (2), and three (3) iterations based on our method, and the cases without the weighted average (BM), with the weighted averages of three A-mode scans (BM(1)) and five A-mode scans (BM(2)) based on the BM-scan technique. The results are obtained by scanning a mirror surface with the unprocessed OCT image shown in the insert at the upper-left corner. The insert at the lower-right corner shows an example of suppression ratio evaluation. The suppression ratio, which is represented by the horizontal (red) line, corresponds to the average of the fluctuating OCT signal intensity ratios along the B-mode scan.

Fig. 6
Fig. 6

Suppression ratios as functions of iteration number in our method based on mirror surface scanning when the process phase shift is equal to the scan phase shift, which is labeled for each curve in the figure.

Fig. 7
Fig. 7

Suppression ratios as functions of iteration number with various scan phase shift values, as labeled for the curves in the figure, based on mirror surface scanning. The results are obtained by using 90 degrees as the process phase shift in our method.

Fig. 8
Fig. 8

Suppression ratios as functions of the tilt angle of the scanned rough surface under various process conditions, including one (1), two (2), and three (3) iterations based on our method, and the BM-scan (BM) technique, when both scan and process phase shifts are 90 degrees.

Fig. 9
Fig. 9

Suppression ratios as functions of the B-mode pixel size based on mirror surface scanning under various process conditions, including one (1), two (2), and three (3) iterations in our method, and the BM-scan (BM) technique, when both scan and process phase shifts are 90 degrees.

Equations (14)

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S n = r n + m n
S n+1 = r n+1 exp( iθ )+ m n+1 exp( iθ ).
r ˜ n = S n S n+1 exp( iθ ) 1exp( i2θ )
m ˜ n = S n S n+1 exp( iθ ) 1exp( i2θ ) .
S n1 = r n1 exp( iθ )+ m n1 exp( iθ ),
r ^ n = S n1 exp( iθ ) S n exp( i2θ ) 1exp( i2θ )
m ^ n = S n1 exp( iθ ) S n exp( i2θ ) 1exp( i2θ ) .
r n r ˜ n + r ^ n 2 = ( S n1 S n+1 )exp( iθ )+ S n [ 1exp( i2θ ) ] 2[ 1exp( i2θ ) ]
m n m ˜ n + m ^ n 2 = ( S n1 S n+1 )exp( iθ )+ S n [ 1exp( i2θ ) ] 2[ 1exp( i2θ ) ] .
r ˜ n = S n i S n+1 2 = r n + r n+1 2 + m n m n+1 2
m ˜ n = S n +i S n+1 2 = r n r n+1 2 + m n + m n+1 2 .
r n = 2 S n +i( S n1 S n+1 ) 4 = r n1 +2 r n + r n+1 4 m n1 2 m n + m n+1 4
m n = 2 S n i( S n1 S n+1 ) 4 = r n1 2 r n + r n+1 4 + m n1 +2 m n + m n+1 4 .
r n = 2 S n +i( S n1 S n+1 ) 4 = 1 16 ( S n2 +4i S n1 +6 S n 4i S n+1 S n+2 ) = 1 16 ( r n2 +4 r n1 +6 r n +4 r n+1 + r n+2 )+ 1 16 ( m n2 4 m n1 +6 m n 4 m n+1 + m n+2 ).

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