Abstract

In this paper, we introduce a new form of cross-sectional, coherence-gated fluorescence imaging, which we term ‘spectral-domain fluorescence coherence tomography’ (SD-FCT). SD-FCT is accomplished by spectrally detecting self-interference of the spontaneous emission of fluorophores located along the axial (depth) dimension of the sample. We have built a first generation SD-FCT system that utilizes two opposing low numerical-aperture objective lenses in an interferometer and an imaging spectrometer for detecting self-interference of fluorescence emitted from a sample. Here, in proof-of-principle experiments we demonstrate cross-sectional profiling of layered fluorescence phantoms. Narrow (a few micrometers FWHM) axial point-spread functions, large ranging depths (a few hundreds of micrometers) and wide fields of view (>1 mm) were measured. Initial results suggest that SD-FCT may be a viable tool for the investigation of semi-transparent and selectively labeled fluorescent samples.

© 2006 Optical Society of America

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  1. 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, 1178-1181 (1991).
    [CrossRef] [PubMed]
  2. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
    [CrossRef]
  3. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
    [CrossRef] [PubMed]
  4. J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
    [CrossRef] [PubMed]
  5. K. H. Drexhage, "Interaction of light with monomolecular dye layers," Prog. Opt. 12, 163-232 (1974).
    [CrossRef]
  6. K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
    [CrossRef]
  7. A. Lambacher and P. Fromherz, "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer," Appl. Phys. A 63, 207-216 (1996).
    [CrossRef]
  8. A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
    [CrossRef]
  9. S. Hell and E. H. K. Stelzer, "Properties of a 4Pi-confocal fluorescence microscope," J. Opt. Soc. Am. A 9, 2159-2166 (1992).
    [CrossRef]
  10. M. G. L. Gustafsson, D. A. Agard and J. W. Sedat, "Sevenfold improvement of axial resolution in 3D widefiled microscopy using two objective lenses," in Three-dimensional Microscopy: Image Acquisition and Processing II, Tony Wilson and Carol J. Cogswell, eds., 2412, 147-156 (1995).
  11. B. Karamata, P. Lambelet, M. Laubscher, R. P. Salathé, and T. Lasser, "Spatially incoherent illumination as a mechanism for cross-talk suppression in wide-field optical coherence tomography," Opt. Lett. 29, 736-738 (2004).
    [CrossRef] [PubMed]
  12. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, "Full range complex spectral optical coherence tomography technique in eye imaging," Opt. Lett. 27, 1415-1417 (2002).
    [CrossRef]
  13. B. Grajciar, M. Pircher, A. Fercher, and R. Leitgeb, "Parallel Fourier domain optical coherence tomography for in vivo measurement of the human eye," Opt. Express 13, 1131-1137 (2005).
    [CrossRef] [PubMed]
  14. C. Dorrer, N. Belabas, J-P Likforman, and M. Joffre, "Spectral resolution and sampling issues in Fourier transform spectral interferometry," J. Opt. Soc. Am. B 17, 1795-1802 (2000).
    [CrossRef]
  15. N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, "In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve," Opt. Express 12, 367-376 (2004).
    [CrossRef] [PubMed]
  16. S. Yun, G. Tearney, B. Bouma, B. Park, and J. de Boer, "High-speed spectral-domain optical coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).
    [CrossRef] [PubMed]
  17. A. Bilenca, A. Desjardins, B. Bouma, and G. Tearney, "Multicanonical Monte-Carlo simulations of light propagation in biological media," Opt. Express 13, 9822-9833 (2005).
    [CrossRef] [PubMed]

2005

2004

2003

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

S. Yun, G. Tearney, B. Bouma, B. Park, and J. de Boer, "High-speed spectral-domain optical coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).
[CrossRef] [PubMed]

2002

2000

1996

A. Lambacher and P. Fromherz, "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer," Appl. Phys. A 63, 207-216 (1996).
[CrossRef]

1994

1992

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, 1178-1181 (1991).
[CrossRef] [PubMed]

1989

K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
[CrossRef]

1974

K. H. Drexhage, "Interaction of light with monomolecular dye layers," Prog. Opt. 12, 163-232 (1974).
[CrossRef]

Belabas, N.

Bilenca, A.

Bouma, B.

Cantor, C. R.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[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, 1178-1181 (1991).
[CrossRef] [PubMed]

Chen, T.

Cnossen, G.

K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
[CrossRef]

Davis, B.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

de Boer, J.

Del Bene, F.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Desjardins, A.

Dorrer, C.

Drabe, K. E.

K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
[CrossRef]

Drexhage, K. H.

K. H. Drexhage, "Interaction of light with monomolecular dye layers," Prog. Opt. 12, 163-232 (1974).
[CrossRef]

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Fercher, A.

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[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, 1415-1417 (2002).
[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, 1178-1181 (1991).
[CrossRef] [PubMed]

Fromherz, P.

A. Lambacher and P. Fromherz, "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer," Appl. Phys. A 63, 207-216 (1996).
[CrossRef]

Fujimoto, J. G.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [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, 1178-1181 (1991).
[CrossRef] [PubMed]

Goldberg, B. B.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Grajciar, B.

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [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, 1178-1181 (1991).
[CrossRef] [PubMed]

Hell, S.

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Huisken, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Ippolito, S. B.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Izatt, J. A.

Joffre, M.

Karamata, B.

Karl, W. C.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Kowalczyk, A.

Lambacher, A.

A. Lambacher and P. Fromherz, "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer," Appl. Phys. A 63, 207-216 (1996).
[CrossRef]

Lambelet, P.

Lasser, T.

Laubscher, M.

Leitgeb, R.

Likforman, J-P

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Moiseev, L. A.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Nassif, N.

Owen, G. M.

Park, B.

Pierce, M.

Pircher, M.

Puliafito, C. 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, 1178-1181 (1991).
[CrossRef] [PubMed]

Salathé, R. P.

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Stelzer, E. H. K.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

S. Hell and E. H. K. Stelzer, "Properties of a 4Pi-confocal fluorescence microscope," J. Opt. Soc. Am. A 9, 2159-2166 (1992).
[CrossRef]

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Swan, A. K.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Swanson, E. A.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [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, 1178-1181 (1991).
[CrossRef] [PubMed]

Swoger, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Tearney, G.

Unlu, M. S.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

Wiersma, D. A.

K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
[CrossRef]

Wittbrodt, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [PubMed]

Wojtkowski, M.

Yun, S.

Appl. Phys. A

A. Lambacher and P. Fromherz, "Fluorescence interference-contrast microscopy on oxidized silicon using a monomolecular dye layer," Appl. Phys. A 63, 207-216 (1996).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. K. Swan, L. A. Moiseev, C. R. Cantor, B. Davis, S. B. Ippolito, W. C. Karl, B. B. Goldberg, M. S. Unlu, "Toward nanometer-scale resolution in fluorescence microscopy using spectral self-interference," IEEE J. Sel. Top. Quantum Electron. 9, 294-300 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

K. E. Drabe, G. Cnossen, and D. A. Wiersma, "Localization of spontaneous emission in front of a mirror," Opt. Commun. 73, 91-95 (1989).
[CrossRef]

Opt. Express

Opt. Lett.

Prog. Opt.

K. H. Drexhage, "Interaction of light with monomolecular dye layers," Prog. Opt. 12, 163-232 (1974).
[CrossRef]

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography - principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, "Optical sectioning deep inside live embryos by selective plane illumination microscopy," Science 305, 1007-1009 (2004).
[CrossRef] [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, 1178-1181 (1991).
[CrossRef] [PubMed]

Other

M. G. L. Gustafsson, D. A. Agard and J. W. Sedat, "Sevenfold improvement of axial resolution in 3D widefiled microscopy using two objective lenses," in Three-dimensional Microscopy: Image Acquisition and Processing II, Tony Wilson and Carol J. Cogswell, eds., 2412, 147-156 (1995).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of a spectral domain fluorescence coherence tomography (SD-FCT) system. (a) Principle of operation - the fluorescent sample is located between two matched, opposing low-NA objectives, near the zero differential path length point (z0 ) of the interferometer and is illuminated with a line focus at the excitation wavelength (green). Self-interference fluorescence from the sample (orange) is detected using an imaging spectrometer. The depth information of each fluorophore is encoded by an interferometric frequency modulation of the emission spectrum. (b) Self-interference fluorescence from the sample is imaged along the transversal dimension, and spectrally resolved in the spectral dimension of the two-dimensional detection array. Inverse discrete Fourier transform (DFT-1) performed on each horizontal CCD line results in the axial (depth) ranging profile of the fluorophore distribution.

Fig. 2.
Fig. 2.

SD-FCT experimental setup (Top view) - Excitation light (green) propagated through a pinhole (PH), a cylindrical lens (LC), a vertical slit (S) and a spherical lens (L1), and subsequently reflected off a dichroic mirror (DM1) to produce a ‘y-z’ plane of illumination at the focal volume of two opposing objectives (O1 and O2). The fluorescent sample (FS) was positioned between O1 and O2. Fluorescence emission (orange) was collected by both lenses, reflected by mirrors (M3 and M4) so as to interfere in a 50/50 non-polarizing beam splitter (NPBS1) and finally directed via an emission filter (EF) and a second 50/50 non-polarizing beam splitter (NPBS2) toward the spectrometer arm (consisting of a diffraction grating (DG), an achromatic lens (LS) and an EMCCD camera (EMCCDS)) and the imaging arm (comprising an imaging lens (L2) and a CCD camera (CCDI)). In addition to directing the excitation beam toward the sample, DM1 filtered back-reflected excitation light, and transmitted fluorescence emission from the right-hand arm of the interferometer to NPBS1. A second dichroic mirror (DM2) was positioned in the left-hand arm of the interferometer to balance dispersion and filter excitation light.

Fig. 3.
Fig. 3.

Ray tracing diagrams of the optical system of Fig. 1 in the direction of the imaging spectrometer path. Top - vertical plane, Bottom - horizontal plane; Green - excitation light, Orange - emission light.

Fig. 4.
Fig. 4.

(a) Axial PSF’s of a single layer of 100 nm fluorescent beads versus axial position of the layer (blue, red, green, magenta, cyan). Also shown is the cooled, electron-multiplying CCD (EMCCDS) read-out noise (black). (b) Typical spectral interferogram signal (red, solid line) and fluorescence emission spectrum (blue, dashed line) measured with a single layer of 100 nm fluorescent beads.

Fig. 5.
Fig. 5.

Detection performance and axial resolution of SD-FCT. (a) SNR dependence of the axial PSF on the axial position of the single fluorescent layer (closed circles). Also shown is the theoretical fit using Eq. (3) (dashed line) (b) Axial resolution at different axial positions of the single fluorescent layer.

Fig. 6.
Fig. 6.

SD-FCT measurement of a fluorescent sample comprising two layers of 100 nm fluorescent beads separated by 120 μm. (a) Schematic of the two-layered fluorescent phantom. (b) Fluorescence emission distribution along the axial (‘z’) and transversal (‘y’) coordinates of the two-layered sample. (c) SD-FCT tomogram of the two-layered fluorescent phantom.

Equations (4)

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

I D ( ω ) = N m = 1 M a ( ζ m ) P m { A m ( t ) } ( ω ) + b ( ζ m ) P m { A m ( t 2 ζ m c ) } ( ω ) 2
= N I F ( ω ) E Z F [ ( a 2 ( ζ F ) + b 2 ( ζ F ) ) ( 1 + V ( ζ F ) cos ( 2 ζ F c ω ) ) ]
N 2 Γ F ( z ) E Z F [ ( a 2 ( ζ F ) + b 2 ( ζ F ) ) ] + E Z F [ ( a 2 ( ζ F ) + b 2 ( ζ F ) ) V ( ζ F ) π 2 ( Γ F ( z ζ F ) + Γ F ( z ζ F ) ) ] 2
SNR Reduction ( z ) = sin c 2 ( z 2 z max ) e π 2 W 2 8 ln 2 ( z z max ) 2

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