Abstract

Phase variance-based motion contrast is demonstrated using two phase analysis methods in a spectral domain optical coherence tomography system. Mobility contrast is demonstrated for an intensity matched Intralipid solution placed without flow within agarose wells. Vasculature oriented transversely to the imaging direction has been imaged for 3-4 dpf in vivo zebrafish using the phase variance contrast methods. 2D phase variance contrast images are demonstrated with imaging times only 25% higher than a Doppler flow image with comparable statistics. En face images created by integrating depth regions of 3D zebrafish intensity and phase variance contrast data demonstrate vasculature consistent with expected images.

© 2007 Optical Society of America

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    [PubMed]
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2006

2005

2004

2003

2001

S. Isogai, M. Horiguchi and B. M. Weinstein, "The vascular anatomy of the developing zebrafish: an atlas of embryonic and early larval development," Developmental Biology 230, 278 (2001).
[CrossRef] [PubMed]

1999

1995

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

1994

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

1986

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Bajraszewski, T.

Berisha, F.

Boppart, S. A

Bouma, B. E.

Bouma, B.E.

Bressler, N. M.

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Bressler, S. B.

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Chen, T. C.

Costanza, M. A.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

de Boer, J. F.

de Boer, J.F.

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239 (2003).
[CrossRef]

W. Drexler, U. Morgner, F. X. Kartner, C. Pitris, S. A Boppart, X. D. Li, E. P. Ippen and J. G. Fujimoto, "In vivo ultrahigh resolution optical coherence tomography," Opt. Lett. 24, 1221 (1999).
[CrossRef]

Elzaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

Fercher, A. F.

R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski and T. Bajraszewski, "Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography," Opt. Lett. 29, 171 (2004).
[CrossRef] [PubMed]

A. F. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239 (2003).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gragoudas, E. S.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Guyer, D. R.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski and T. Bajraszewski, "Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography," Opt. Lett. 29, 171 (2004).
[CrossRef] [PubMed]

A. F. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239 (2003).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

Hong, Y.

Hope-Ross, M.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Horiguchi, M.

S. Isogai, M. Horiguchi and B. M. Weinstein, "The vascular anatomy of the developing zebrafish: an atlas of embryonic and early larval development," Developmental Biology 230, 278 (2001).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Ippen, E. P.

Isogai, S.

S. Isogai, M. Horiguchi and B. M. Weinstein, "The vascular anatomy of the developing zebrafish: an atlas of embryonic and early larval development," Developmental Biology 230, 278 (2001).
[CrossRef] [PubMed]

Izatt, J. A.

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

Kartner, F. X.

Krupsky, S.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239 (2003).
[CrossRef]

Leitgeb, R. A.

Li, X. D.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Maguire, M. G.

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Makita, S.

Morgner, U.

Mujat, M.

Nassif, N.

Orlock, D. A.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Park, B. H.

Park, B.H.

Pierce, M. C.

Pierce, M.C

Pitris, C.

Puliafito, C. A.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Rohrer, K. T.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Sarunic, M. V.

Schachat, A. P.

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Schmetterer, L.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Shields, W.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Slakter, J. S.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Sobel, R. S.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Sorenson, J. A.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Sykes, S. O.

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Tearney, G. J.

Tearney, G.J.

Tindel, L. J.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Vakoc, B.

Weinstein, B. M.

S. Isogai, M. Horiguchi and B. M. Weinstein, "The vascular anatomy of the developing zebrafish: an atlas of embryonic and early larval development," Developmental Biology 230, 278 (2001).
[CrossRef] [PubMed]

White, B. R.

Wojtkowski, M.

Yamanari, M.

Yang, C. H.

Yannuzzi, L. A.

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

Yun, S. H.

Yun, S.H.

Zang, E.

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

Arch. Ophthalmology

S. O. Sykes, N. M. Bressler, M. G. Maguire, A. P. Schachat and S. B. Bressler, "Detecting recurrent choroidal neovascularization. Comparison of clinical examination with and without fluorescein angiography," Arch. Ophthalmology 112, 1561 (1994).
[CrossRef]

Developmental Biology

S. Isogai, M. Horiguchi and B. M. Weinstein, "The vascular anatomy of the developing zebrafish: an atlas of embryonic and early larval development," Developmental Biology 230, 278 (2001).
[CrossRef] [PubMed]

Ophthalmology

L. A. Yannuzzi, K. T. Rohrer, L. J. Tindel, R. S. Sobel, M. A. Costanza, W. Shields and E. Zang, "Fluorescein angiography complication survey," Ophthalmology 93, 611 (1986).
[PubMed]

M. Hope-Ross, L. A. Yannuzzi, E. S. Gragoudas, D. R. Guyer, J. S. Slakter, J. A. Sorenson, S. Krupsky, D. A. Orlock and C. A. Puliafito, "Adverse reactions to indocyanine green," Ophthalmology 101, 529 (1994).
[PubMed]

Opt. Commun.

A. F. Fercher, C. K. Hitzenberger, G. Kamp and S. Y. Elzaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 117, 43 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239 (2003).
[CrossRef]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito and J. G Fujimoto, "Optical coherence tomography," Science 254, 1178 (1991).
[CrossRef] [PubMed]

Other

J. D. Gass, "Stereoscopic atlas of macular diseases," 4th ed., (Mosby, 1997).

"The Zebrafish Information Network," www.zfin.org.

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Figures (10)

Fig. 1.
Fig. 1.

Calculated phase changes between OCT A-scans of paper sample for approximately 1.5 radians of bulk motion. (a) Histogram of phase changes for all depths of the sample region, independent of intensity. (b) Histogram of phase changes for reflections with OCT signal at least 15dB higher than the calculated noise level. The standard deviation calculated for the phase changes in (a) and (b) are 1.27 radians and 0.31 radians, respectively. (c) Standard deviation of the phase changes plotted against the OCT intensity threshold level applied.

Fig. 2.
Fig. 2.

Schematic of phase analysis technique used to produce phase variance for contrast analysis. PhC: phase conditioning to limit maximum phase change between measurements due to phase wrapping.

Fig. 3.
Fig. 3.

Schematic of transverse scan patterns for (a) MB scan and (b) BM scan

Fig. 4.
Fig. 4.

Cross-sectional images of 2% agarose wells filled with OCT intensity-matched 0.1% Intralipid imaged using MB scan. (a) Schematic cross sectional image of filled agarose wells. (b) Averaged OCT intensity image showing limited contrast between Intralipid and agarose regions. (c) Phase variance contrast image demonstrating phase motion of Intralipid without any induced flow. The variance contrast image uses a threshold based on phase information to eliminate high phase noise terms from the image. Data shown uses 200 transverse pixels with an image size is 1.6mm x 1.0mm.

Fig. 5.
Fig. 5.

Phase error calculated as a function of normalized OCT interferometer signal data. Noise estimation (red straight line) was performed by fitting data in the 10dB to 25dB range to the expected form of SNR-limited phase error.

Fig. 6.
Fig. 6.

Images of 3dpf zebrafish (a) Brightfield confocal image. (b) Confocal image of GFP-labelled vasculature. (c) Histology of zebrafish corresponding to a slice location depicted in (a) and (b). Dorsal side of fish oriented towards the top of each images.

Fig. 7.
Fig. 7.

MB-scan images of 4 dpf zebrafish tail. (a) OCT intensity averaged image. (b) Phase variance contrast image using T2=1ms and T1=40μs. Conventional Doppler OCT technique images average phase change between successive A-scans averaged over (c) 10 and (d) 100 total scans per transverse position. The image scale used for (c) and (d) is +/- 0.12 radians, which corresponds to +/- 200 μm/s. Arrows in all motion contrast images depict the locations of phase contrast identified in (b). Dorsal side of fish is identified on the left-hand side of images. Data shown uses 200 transverse pixels with an image size is 480μm x 280μm.

Fig. 8.
Fig. 8.

MB-scan phase variance contrast images of 4dpf zebrafish tail. The contrast images are created from the same phase variance data (a) without the removal of SNR-limited phase noise and (b) with the phase noise removal, reproduced from Fig. 7(b). Threshold on images is set at SNR=1 for OCT intensity. Data shown uses 200 transverse pixels with an image size is 480μm x 280μm.

Fig. 9.
Fig. 9.

BM-Scan images of 4 dpf zebrafish tail over the same transverse scan region as used for Fig. 7. (a) OCT averaged intensity image. (b) Phase contrast image with T2= 10ms and numerical removal of phase noise. Rank 1 median filters were applied horizontally and vertically to contrast image. Dorsal side of fish is aligned to left hand side of images. Locations of motion contrast indicated with arrows on both images. Data shown uses 200 transverse pixels with an image size is 480μm x 280μm.

Fig. 10.
Fig. 10.

En face images over zebrafish heart. (a) OCT intensity summation logarithmic image (b) Phase variance summation logarithmic image. (c) Confocal image of GFP-labeled 3 dpf zebrafish.

Equations (46)

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v axial ( z ) = λ 4 πnT ϕ ( z , t + T ) ϕ ( z , t ) = λ 4 πnT Δϕ ( T )
Δϕ ( z , T ) = Δ ϕ scatterer ( z , T ) + Δ ϕ bulk ( T ) + Δ ϕ SNR ( z ) + Δ ϕ error , other ( T )
σ Δϕ , SNR ( z ) = 1 SNR OCT ( z )
Δ ϕ bulk ( T ) = z [ I ( z ) Δϕ ( z , T ) ] z [ I ( z ) ]
σ Δϕ 2 ( z , T ) = σ Δϕ , scatterer 2 ( z , T ) + σ error , bulk 2 + σ Δϕ , SNR 2 ( z ) + σ error , other 2 ( z )
σ Δϕ , SNR 2 ( z ) = 1 SNR OCT ( z ) = < N 2 > S ( z ) 2
Ĩ = I exp ( i ϕ ) = S exp ( i ϕ S ) + N exp ( i ϕ N )
Ĩ 2 = I 2 = S 2 + N 2 2 SN cos ( ϕ S ϕ N )
F ( k ) = j 2 S j ( k ) cos ( k z j + ϕ S j ) + F D C ( k ) + N ( k )
Ĩ ( z ) = j FT ( 2 S j ( k ) cos ( k z j + ϕ S j ) ) + FT ( F D C ( k ) ) + FT ( N ( k ) )
Ĩ ( z ) = I ( z ) exp ( i ϕ ( z ) ) = FT ( F ( k ) ) = k M F ( k ) exp ( ikz )
= I Re ( z ) + i I Im ( z ) = k M F ( k ) cos ( k z ) i k M F ( k ) sin ( k z )
k M cos ( k z ) cos ( k z j ) = k M sin ( k z ) sin ( k z j ) = M 2 δ ( z z j ) , k M cos ( k z ) sin ( k z j ) = 0
FT ( F DC ( k ) ) = k M F D C ( k ) exp ( ikz ) δ ( z ) k M F D C ( k ) = δ ( z ) η τ h v k M P R ( k )
= δ ( z ) η P R τ h v
S ˜ ( z ) = k M j 2 S j ( k ) ( cos ( k z j ) cos ( ϕ S j ) sin ( k z j ) sin ( ϕ S j ) ) exp ( ikz )
S ˜ ( z ) = k M j 2 S j ( k ) ( cos ( k z j ) cos ( k z ) cos ( ϕ S j ) + i sin ( k z j ) sin ( k z ) sin ( ϕ S j ) )
= k M j = 1 S j ( k ) ( cos ( ϕ S j ) + i sin ( ϕ S j ) ) δ ( z z j ) = j exp ( i ϕ S j ) k M S j ( k ) δ ( z z j )
= η τ h v j exp ( i ϕ S j ) k M P R ( k ) P S ( k ) R j δ ( z z j )
= η P R P S τ h v j exp ( i ϕ S j ) R j δ ( z z j ) = S ( z ) exp ( i ϕ S ( z ) )
FT ( N ( k ) ) = N ˜ ( z ) = N ( z ) exp ( i ϕ N ( z ) )
z M f ˜ ( z ) 2 = z M f ˜ ( z ) ( f ˜ ( z ) ) * = z M f ˜ ( z ) k M f ( k ) exp ( ikz )
= k M f ( k ) z M f ˜ ( z ) exp ( ikz ) = z M k M f ( k ) exp ( ikz ) k M f ( k ) exp ( ik′z )
= k M f ( k ) k M f ( k ) z M exp ( i ( k k ) z )
= M k M f ( k ) k M f ( k ) δ ( k k ) = M k M f ( k ) 2
z M N ˜ ( z ) 2 = M k M N ( k ) 2
z M N ˜ ( z ) 2 = M N ˜ ( z ) 2 z = M k M N ( k ) 2 = M 2 N ( k ) 2 k = M 2 σ N ( k ) 2 k
N ˜ ( z ) 2 z = M σ 2 N ( k ) k = η τ h v M P R ( k ) k = η P R τ h v
N ˜ ( z ) = N ( z ) exp ( i ϕ N ( z ) ) = N Re ( z ) + i N Im ( z )
N ˜ ( z ) 2 = N Re ( z ) 2 + N Im ( z ) 2
N ˜ ( z ) 2 = N Re ( z ) 2 + N Im ( z ) 2 = σ N Re 2 + σ N Im 2
σ N Re 2 = σ N Im 2 = 1 2 N ˜ ( z ) 2 = η P R τ 2 h v
P ( N Re ) exp ( N Re 2 / 2 σ Nre 2 )
P ( N ) = 0 N P ( N Re ) P ( N Im = N 2 N Re 2 ) d N Re
= 0 N P o 2 exp ( ( N 2 ) / 2 σ Nre 2 ) d N Re
P ( N ) = 2 < N ˜ ( z ) 2 > N exp ( ( N 2 ) / < N ˜ ( z ) 2 > )
σ N 2 = N ˜ ( z ) 2 = η P R τ h v
Ĩ ( z ) = I ( z ) exp ( i ϕ ( z ) ) = FT ( F ( k ) ) = S ˜ ( z ) + N ˜ ( z )
= S ( z ) exp ( i ϕ S ( z ) ) + N ( z ) exp ( i ϕ N ( z ) )
Ĩ ( z ) 2 = I ( z ) 2 = S ( z ) 2 + N ( z ) 2 2 S ( z ) N ( z ) cos ( ϕ S ( z ) ϕ N ( z ) )
Ĩ ( z ) 2 = S ( z ) 2 + N ( z ) 2
S NR = S ( z , R = 1 ) 2 σ N 2 = S 2 N ( z ) 2 = ( ( η τ h v ) P R P S ) 2 ( η τ h v ) P R = η P S τ h v
tan ( ϕ ( z ) ) = N Im ( z ) S ( z ) + N Re ( z ) , ϕ ( z ) N Im ( z ) S ( z )
P ( ϕ ( z ) ) exp ( ϕ ( z ) 2 S ( z ) 2 / N ( z ) 2 )
σ ϕ 2 ( z ) = N ˜ ( z ) 2 2 S ( z ) 2
ϕ Δ ϕ 2 ( z ) = 2 σ ϕ 2 ( z ) = N ˜ ( z ) 2 S ( z ) 2 = 1 SNR ( z )

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