Abstract

We have developed a new theoretical description of the optical coherence tomography (OCT) technique for imaging in highly scattering tissue. The description is based on the extended Huygens–Fresnel principle, valid in both the single- and multiple-scattering regimes. The so-called shower curtain effect, which manifests itself in a standard OCT system, is an inherent property of the present theory. We demonstrate that the shower curtain effect leads to a strong increase in the heterodyne signal in a standard OCT system. This is in contrast to previous OCT models, where the shower curtain effect was not taken into account. The theoretical analysis is verified by measurements on samples consisting of aqueous suspensions of microspheres. Finally, we discuss the use of our new theoretical model for optimization of the OCT system.

© 2000 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
    [CrossRef] [PubMed]
  3. J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
    [CrossRef]
  4. M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
    [CrossRef] [PubMed]
  5. M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Contrast and resolution in the optical coherence microscopy of dense biological tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and other Diseases II, R. R. Alfano, ed., Proc. SPIE2387, 193–203 (1995).
    [CrossRef]
  6. Y. Pan, R. Birngruber, R. Engelhardt, “Contrast limits of coherence-gated imaging in scattering media,” Appl. Opt. 36, 2979–2983 (1997).
    [CrossRef] [PubMed]
  7. J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
    [CrossRef]
  8. L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.
  9. R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  10. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [CrossRef]
  11. M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [CrossRef] [PubMed]
  12. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 16.1.
  13. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  14. H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
    [CrossRef]
  15. H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  16. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  17. Note that the (algebraically simple) results given below are, strictly speaking, valid only for propagation geometries where A=D, as is obtained in the case of interest.
  18. I. Dror, A. Sandrov, N. S. Kopeika, “Experimental investigation of the influence of the relative position of the scattering layer on image quality: the shower curtain effect,” Appl. Opt. 37, 6495–6499 (1998).
    [CrossRef]
  19. D. A. Boas, K. K. Bizheva, A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23, 319–321 (1998).
    [CrossRef]
  20. R. F. Lutomirski, “Atmospheric degradation of electrooptical system performance,” Appl. Opt. 17, 3915–3921 (1978).
    [CrossRef] [PubMed]

1998

1997

1996

1995

1993

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1989

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

1987

1979

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

1978

1971

Andersen, P. E.

L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.

Birngruber, R.

Bizheva, K. K.

Boas, D. A.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Bonner, R. F.

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Contrast and resolution in the optical coherence microscopy of dense biological tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and other Diseases II, R. R. Alfano, ed., Proc. SPIE2387, 193–203 (1995).
[CrossRef]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Dror, I.

Engelhardt, R.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gandjbakhche, A. S.

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hanson, S. G.

H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jacques, S. L.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Knüttel, A.

J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[CrossRef]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

Kopeika, N. S.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lutomirski, R. F.

Pan, Y.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sandrov, A.

Schmitt, J. M.

J. M. Schmitt, A. Knüttel, “Model of optical coherence tomography of heterogeneous tissue,” J. Opt. Soc. Am. A 14, 1231–1242 (1997).
[CrossRef]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Contrast and resolution in the optical coherence microscopy of dense biological tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and other Diseases II, R. R. Alfano, ed., Proc. SPIE2387, 193–203 (1995).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Siegel, A. M.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 16.1.

Star, W. M.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Thrane, L.

L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.

Van Gemert, M. J. C.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Yadlowsky, M. J.

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Multiple scattering in optical coherence microscopy,” Appl. Opt. 34, 5699–5707 (1995).
[CrossRef] [PubMed]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Contrast and resolution in the optical coherence microscopy of dense biological tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and other Diseases II, R. R. Alfano, ed., Proc. SPIE2387, 193–203 (1995).
[CrossRef]

Yura, H. T.

H. T. Yura, S. G. Hanson, “Effects of receiver optics contamination on the performance of laser velocimeter systems,” J. Opt. Soc. Am. A 13, 1891–1902 (1996).
[CrossRef]

H. T. Yura, S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

R. F. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.

Appl. Opt.

IEEE Trans. Biomed. Eng.

M. J. C. Van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Acta

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[CrossRef]

Opt. Lett.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other

J. M. Schmitt, A. Knüttel, A. S. Gandjbakhche, R. F. Bonner, “Optical characterization of dense tissues using low-coherence interferometry,” in Holography, Interferometry, and Optical Pattern Recognition in Biomedicine III, H. Podbielska, ed., Proc. SPIE1889, 197–211 (1993).
[CrossRef]

M. J. Yadlowsky, J. M. Schmitt, R. F. Bonner, “Contrast and resolution in the optical coherence microscopy of dense biological tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and other Diseases II, R. R. Alfano, ed., Proc. SPIE2387, 193–203 (1995).
[CrossRef]

L. Thrane, H. T. Yura, S. G. Hanson, P. E. Andersen, “Optical coherence tomography of heterogeneous tissue: calculation of the heterodyne signal,” in Conference on Lasers and Electro-Optics (CLEO/Europe ’98), OSA 1998 Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 58.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 16.1.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Note that the (algebraically simple) results given below are, strictly speaking, valid only for propagation geometries where A=D, as is obtained in the case of interest.

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Figures (5)

Fig. 1
Fig. 1

Sample arm geometry of the OCT system.

Fig. 2
Fig. 2

Heterodyne efficiency factor Ψ as a function of depth z for diffuse backscattering with the shower curtain effect included (curve 1) and for specular reflection (curve 3). Curve 2 is calculated for diffuse backscattering but with the shower curtain effect excluded, and curve 4 is the case of pure single (back-) scattering (λ=814 nm, μs=20 mm-1, g=0.955 (θrms=0.3 rad), n=1.4, f=5 mm, w0=0.5 mm).

Fig. 3
Fig. 3

Theoretical curves and measurements of the heterodyne efficiency factor Ψ as a function of the scattering coefficient μs for the case of diffuse backscattering (dashed curve, squares) and specular reflection (solid curve, circles) at the discontinuity. The probing depth z is 0.5 mm, and the typical standard deviation for the squares is ±7.8% [λ=814 nm, g=0.929 (θrms=0.38 rad), n=1.33, f=16 mm, w0=0.125 mm, δ=1.256 mm, ng=1.5].

Fig. 4
Fig. 4

Heterodyne efficiency factor Ψ as a function of the focal length f or the numerical aperture NA when the probing depth z=0.5 mm and the shower curtain effect is included. The dashed and the solid curves are calculated for the case of a low (2-mm-1) and a high (10-mm-1) scattering coefficient, respectively [λ=814 nm, g=0.955 (θrms=0.3 rad), n=1.4, w0=0.5 mm].

Fig. 5
Fig. 5

Mean square heterodyne signal current i2 as a function of the focal length f or the numerical aperture NA, with (solid curve) and without (dashed curve) the shower curtain effect. The probing depth z=0.5 mm [λ=814 nm, g=0.955 (θrms=0.3 rad), n=1.4, w0=0.5 mm]. (a) μs=2 mm-1, (b) μs=10 mm-1.

Equations (35)

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

i(z)ReUR(p, t)US*(p, t+τ)dp,
i(z)|g(τ) ReUR(p)US*(p)dp,
i2(z)=2α2|g(τ)|2×ReΓS(p1, p2; z)ΓR(p1, p2)dp1dp2,
ΓR(p1, p2)=UR(p1)UR*(p2),
ΓS(p1, p2; z)=US(p1; z)US*(p2; z)
UR(p, t)=PRπw02 exp-p221w02+ikf×exp[iωRt+φR(t)],
USi(p, t)=PSπw02 exp-p221w02+ikfexp[iωSt],
US(p; z)=UB(r; z)G(r, p; z)dr,
ΓS(p1, p2; z)=UB(r1; z)UB*(r2; z)×G(r1, p1; z)G*(r2, p2; z)dr1dr2.
UB(r1)UB*(r2)G(r1, p1)G*(r2, p2)=UB(r1)UB*(r2)G(r1, p1)G*(r2, p2).
UB(r1)UB*(r2)=4πk2δ(r1-r2)IB(r1),
IB(r)=RdPSπexp(-μsz)exp(-r2/wH2)wH2+[1-exp(-μsz)]exp(-r2/wS2)wS2.
wH2=w02A-Bf2+Bkw02,
wS2=w02A-Bf2+Bkw02+2Bkρ02,
ΓS(p1, p2)=4πk2IB(r)G(r, p1)G*(r, p2)dr.
G(r, p1)G*(r, p2)=G0(r, p1)G0*(r, p2)Γpt(ρ),
G0(r, p)=-ik2πBbexp-ik2Bb(Abr2-2r·p+Dbp2),
Γpt=exp{i[ϕ(p1)-ϕ(p2)]}=exp{-σ2[1-bϕ(ρ)]},
bϕ(ρ)=exp(-ρ2/ρϕ2),
Γpt(ρ)exp(-μsz)+[1-exp(-μsz)]exp(-ρ2/ρ02)
ρ02=ρϕ2/μsz.
i2(z)=2α2PRPSσbπ2exp(-μsz)exp(-r2/wH2)wH2+(1-exp(-μsz))exp(-r2/wS2)wS22dr,
i2(z)=α2PRPSσbπwH2exp(-2μsz)+2 exp(-μsz)[1-exp(-μsz)]1+wS2/wH2+[1-exp(-μsz)]2wH2wS2i20Ψ(z).
wH2=fkw02;wH2wS2=11+2w0ρ0(z)2.
ρ0(z)=3μszλπθrms1+nd(z)z,
Ψ(z)=exp(-2μsz)+[1-exp(-2μsz)]wH2wS2,
ρ0(z)=12μszλπθrms.
d(z)=f-(z/n)+δ(1-1/ng),
Γpt(ρ)=exp-ρ240Lk2θrms2(s)μs(s)B(s)B(L)2ds,
B(s)=sn,0sz,
B(L)=Bb=B=d+z/n.
μs(s)=μs,0sz0,z<sL,
θrms(s)=θrms,0sz0,z<sL.
Γpt(ρ)=exp-ρ2π2θrms2μsz3λ2z2(nd+z)2.
ρ0(z)=3μszλπθrms1+nd(z)z.

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