Abstract

We have performed an experiment of the scattering of the near field on a prism created by a laser wave, evanescent wave (EW), or plane wave (PW) of an incident angle slightly larger than or smaller than the critical angle, by a thin fiber of subwavelength diameter set above the prism, and we made an analytical theory of an adapted model for the experiment. We have been able to analyze the experimental data exactly by the model theory better than any other theory we have ever known. The importance of the multiple interaction of the wave between the fiber and the surface and also the close similarity of the scattering characteristics between the EW and the PW mentioned above have been acknowledged by the analysis of the data obtained.

© 2012 Optical Society of America

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References

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  1. A. Sommerfeld, Vorlesungen ueber Theoretische Physik IV, Optik, 3 Aufl. (Verlag Harri Deutsch, 1989), Sect. 5.
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 11.4.
  3. P. Leurgans and A. F. Turner, “Frustrated total reflection interference filters,” J. Opt. Soc. Am. 37, 983 (1947).
  4. P. A. Temple, “Total internal reflection microscopy: a surface inspection technique,” Appl. Opt. 20, 2656–2664 (1981).
    [CrossRef]
  5. D. Axelrod, “Total internal reflection fluorescent microscopy,” J. Microsc. 129, 19–28 (1983).
    [CrossRef]
  6. L. Novotny, “The history of near-field optics,” Progress in Optics, Vol. 50, E. Wolf, ed. (Elsevier, 2007), Chap. 5.
  7. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
    [CrossRef]
  8. B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
    [CrossRef]
  9. B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
    [CrossRef]
  10. J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
    [CrossRef]
  11. C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
    [CrossRef]
  12. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), Chap. 15.
  13. G. Videen and D. Ngo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70–78 (1997).
    [CrossRef]
  14. A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
    [CrossRef]
  15. L. Helden, E. Eremina, N. Riefler, C. Hertlein, C. Bechinger, Y. Eremin, and T. Wriedt, “Single-particle evanescent light scattering simulations for total internal reflection microscopy,” Appl. Opt. 45, 7299–7308 (2006).
    [CrossRef]
  16. E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
    [CrossRef]
  17. P. A. Martin, Multiple Scattering Interaction of Time-Harmonic Waves with N Obstacles (Cambridge University, 2006).
  18. H. A. Yousif and S. Köhler, “Scattering by two penetrable cylinders at oblique incidence. II. Numerical examples,” J. Opt. Soc. Am. A 5, 1097–1104 (1988).
    [CrossRef]
  19. F. Tajima and Y. Nishiyama, “Multiple scattering effect in the Young-like interference pattern of an optical wave scattered by a double cylinder,” Opt. Rev. 15, 75–83 (2008).
    [CrossRef]
  20. F. Tajima and Y. Nishiyama, “Light scattering from a birefringent cylinder, spider silk, slimmer than the wavelength approaches dipole radiation,” J. Opt. Soc. Am. A 22, 1127–1131(2005).
    [CrossRef]
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965), Sect. 9.1.79.

2010 (1)

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

2008 (1)

F. Tajima and Y. Nishiyama, “Multiple scattering effect in the Young-like interference pattern of an optical wave scattered by a double cylinder,” Opt. Rev. 15, 75–83 (2008).
[CrossRef]

2006 (1)

2005 (2)

2001 (1)

A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
[CrossRef]

2000 (1)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

1997 (3)

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

G. Videen and D. Ngo, “Light scattering from a cylinder near a plane interface: theory and comparison with experimental data,” J. Opt. Soc. Am. A 14, 70–78 (1997).
[CrossRef]

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

1988 (1)

1984 (1)

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[CrossRef]

1983 (1)

D. Axelrod, “Total internal reflection fluorescent microscopy,” J. Microsc. 129, 19–28 (1983).
[CrossRef]

1981 (1)

1947 (1)

P. Leurgans and A. F. Turner, “Frustrated total reflection interference filters,” J. Opt. Soc. Am. 37, 983 (1947).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965), Sect. 9.1.79.

Axelrod, D.

D. Axelrod, “Total internal reflection fluorescent microscopy,” J. Microsc. 129, 19–28 (1983).
[CrossRef]

Bechinger, C.

Bielefeldt, H.

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 11.4.

Carminati, R.

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Deckert, V.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Denk, W.

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[CrossRef]

Doicu, A.

A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
[CrossRef]

Eremin, Y.

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

L. Helden, E. Eremina, N. Riefler, C. Hertlein, C. Bechinger, Y. Eremin, and T. Wriedt, “Single-particle evanescent light scattering simulations for total internal reflection microscopy,” Appl. Opt. 45, 7299–7308 (2006).
[CrossRef]

A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
[CrossRef]

Eremina, E.

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

L. Helden, E. Eremina, N. Riefler, C. Hertlein, C. Bechinger, Y. Eremin, and T. Wriedt, “Single-particle evanescent light scattering simulations for total internal reflection microscopy,” Appl. Opt. 45, 7299–7308 (2006).
[CrossRef]

Girard, C.

C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
[CrossRef]

Greffet, J. J.

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Grishina, N.

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

Hecht, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), Chap. 15.

Helden, L.

Hertlein, C.

Inouye, Y.

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

Köhler, S.

Lanz, M.

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[CrossRef]

Leurgans, P.

P. Leurgans and A. F. Turner, “Frustrated total reflection interference filters,” J. Opt. Soc. Am. 37, 983 (1947).

Martin, O. J. F.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Martin, P. A.

P. A. Martin, Multiple Scattering Interaction of Time-Harmonic Waves with N Obstacles (Cambridge University, 2006).

Ngo, D.

Nishiyama, Y.

F. Tajima and Y. Nishiyama, “Multiple scattering effect in the Young-like interference pattern of an optical wave scattered by a double cylinder,” Opt. Rev. 15, 75–83 (2008).
[CrossRef]

F. Tajima and Y. Nishiyama, “Light scattering from a birefringent cylinder, spider silk, slimmer than the wavelength approaches dipole radiation,” J. Opt. Soc. Am. A 22, 1127–1131(2005).
[CrossRef]

Novotny, L.

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

L. Novotny, “The history of near-field optics,” Progress in Optics, Vol. 50, E. Wolf, ed. (Elsevier, 2007), Chap. 5.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), Chap. 15.

Pohl, D. W.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[CrossRef]

Riefler, N.

Sick, B.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Vorlesungen ueber Theoretische Physik IV, Optik, 3 Aufl. (Verlag Harri Deutsch, 1989), Sect. 5.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965), Sect. 9.1.79.

Tajima, F.

F. Tajima and Y. Nishiyama, “Multiple scattering effect in the Young-like interference pattern of an optical wave scattered by a double cylinder,” Opt. Rev. 15, 75–83 (2008).
[CrossRef]

F. Tajima and Y. Nishiyama, “Light scattering from a birefringent cylinder, spider silk, slimmer than the wavelength approaches dipole radiation,” J. Opt. Soc. Am. A 22, 1127–1131(2005).
[CrossRef]

Temple, P. A.

Turner, A. F.

P. Leurgans and A. F. Turner, “Frustrated total reflection interference filters,” J. Opt. Soc. Am. 37, 983 (1947).

Videen, G.

Wild, U. P.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 11.4.

Wriedt, T.

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

L. Helden, E. Eremina, N. Riefler, C. Hertlein, C. Bechinger, Y. Eremin, and T. Wriedt, “Single-particle evanescent light scattering simulations for total internal reflection microscopy,” Appl. Opt. 45, 7299–7308 (2006).
[CrossRef]

A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
[CrossRef]

Yousif, H. A.

Zenobi, R.

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984).
[CrossRef]

Comput. Phys. Commum. (1)

A. Doicu, Y. Eremin, and T. Wriedt, “Scattering of evanescent wave by a particle on or near a plane surface,” Comput. Phys. Commum. 134, 1–10 (2001).
[CrossRef]

J. Appl. Phys. (1)

B. Hecht, H. Bielefeldt, Y. Inouye, D. W. Pohl, and L. Novotny, “Facts and artifacts in near-field optical microscopy,” J. Appl. Phys. 81, 2492–2498(1997).
[CrossRef]

J. Chem. Phys. (1)

B. Hecht, B. Sick, U. P. Wild, V. Deckert, R. Zenobi, O. J. F. Martin, and D. W. Pohl, “Scanning near-field optical microscopy with aperture probes: fundamentals and applications,” J. Chem. Phys. 112, 7761–7774 (2000).
[CrossRef]

J. Microsc. (1)

D. Axelrod, “Total internal reflection fluorescent microscopy,” J. Microsc. 129, 19–28 (1983).
[CrossRef]

J. Opt. (1)

E. Eremina, Y. Eremin, N. Grishina, and T. Wriedt, “Total internal reflection microscopy: examination of competitive schemes via discrete sources method,” J. Opt. 12, 095703 (2010).
[CrossRef]

J. Opt. Soc. Am. (1)

P. Leurgans and A. F. Turner, “Frustrated total reflection interference filters,” J. Opt. Soc. Am. 37, 983 (1947).

J. Opt. Soc. Am. A (3)

Opt. Rev. (1)

F. Tajima and Y. Nishiyama, “Multiple scattering effect in the Young-like interference pattern of an optical wave scattered by a double cylinder,” Opt. Rev. 15, 75–83 (2008).
[CrossRef]

Prog. Surf. Sci. (1)

J. J. Greffet and R. Carminati, “Image formation in near-field optics,” Prog. Surf. Sci. 56, 133–237 (1997).
[CrossRef]

Rep. Prog. Phys. (1)

C. Girard, “Near fields in nanostructures,” Rep. Prog. Phys. 68, 1883–1933 (2005).
[CrossRef]

Other (6)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), Chap. 15.

L. Novotny, “The history of near-field optics,” Progress in Optics, Vol. 50, E. Wolf, ed. (Elsevier, 2007), Chap. 5.

A. Sommerfeld, Vorlesungen ueber Theoretische Physik IV, Optik, 3 Aufl. (Verlag Harri Deutsch, 1989), Sect. 5.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 11.4.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965), Sect. 9.1.79.

P. A. Martin, Multiple Scattering Interaction of Time-Harmonic Waves with N Obstacles (Cambridge University, 2006).

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Figures (8)

Fig. 1.
Fig. 1.

Schematic diagram of the experimental setup of a fiber set on base fibers on a prism and the incident and scattered waves.

Fig. 2.
Fig. 2.

(a) Optical micrograph of a fiber on a base fiber at the right-hand side above the prism with h=300nm. (b) Optical micrograph of a fiber on the prism with h=0nm.

Fig. 3.
Fig. 3.

Photograph of scattering patterns for fiber B: (a) EW, (b) PW.

Fig. 4.
Fig. 4.

Ray tracing of PW and bright spots.

Fig. 5.
Fig. 5.

Coordinate system of the bicylinder model: P, observation point; S, prism surface.

Fig. 6.
Fig. 6.

Angular distribution of scattering intensity for λ1 and h=300nm: (a) EW, (b) PW. Circles and squares are measured data. Dotted lines and solid ones show optimum theoretical curves.

Fig. 7.
Fig. 7.

Same as Fig. 6 for λ2.

Fig. 8.
Fig. 8.

Same as Fig. 6 for λ1 and h=0nm.

Tables (2)

Tables Icon

Table 1. Optimum Values with Uncertainties of the Parameters of the Fiber in Air

Tables Icon

Table 2. Optimum Values of Parameters ϕin, n1, D1(=2r10), h and UI

Equations (25)

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

E=E0exp(κy)exp(ikxx)(ϕin>ϕc),
E=E0exp(ikyy)exp(ikxx)(ϕin<ϕc),
E=E0exp[ikrsin(ϕ1+iβ)]=E0mexp(mβ)exp(imϕ1)Jm(kr1).
E=E0exp[ikrsin(ϕ1+α)]=E0mexp(imα)exp(imϕ1)Jm(kr1),
exp(β)=n2sinϕin[(n2sinϕin)21]1/2,
exp(iα)=n2sinϕin+i[1(n2sinϕin)2]1/2.
E1sc(r,ϕ)=m=exp(βm)B1(m)Hm(1)(kr)exp(imϕ)(2πkr)1/2exp[i(krπ/4)]×m=exp(βm)B1(m)exp[im(ϕπ/2)],
B1(m)=nJm(kr10)Jm(n1kr10)Jm(kr10)Jm(n1kr10)n1Hm(1)(kr10)Jm(n1kr10)+Hm(1)(kr10)Jm(n1kr10).
E1sc(r2,ϕ2)=mexp(βm)B1(m)Hm+(1)(kL)J(kr2)exp(iϕ2).
A()H(2)(n2kr20)=J(kr20)+B2()H(1)(kr20),n2A()H(2)(n2kr20)=J(kr20)+B2()H(1)(kr20).
B2()=n2J(kr20)H(2)(n2kr20)J(kr20)H(2)(n2kr20)n2H(1)(kr20)H(2)(n2kr20)+H(1)(kr20)H(2)(n2kr20).
E2sc(r2,ϕ2)=mexp(βm)B1(m)Hm+(1)(kL)B2()H(1)(kr2)exp(iϕ2).
E2sc(r,ϕ)(2πkr)1/2exp[i(krπ/4+kLcosϕ)]×mexp(βm)B1(m)Hm+(1)(kL)B2()exp[i(ϕπ/2)].
E2sc(r1,ϕ1)=mexp(βm)B1(m)m1Hm+m1(1)(kL)B2(m1)×m2Hm1m2(1)(kL)Jm2(kr1)exp(im2ϕ1),
E3sc(r1,ϕ1)=mexp(βm)B1(m)m1Hm+m1(1)(kL)B2(m1)×m2Hm1m2(1)(kL)B1(m2)Hm2(1)(kr1)exp(im2ϕ1).
E3sc(r,ϕ)(2πkr)1/2exp[i(krπ/4)]mexp(βm)B1(m)m1Hm+m1(1)(kL)B2(m1)×m2Hm1m2(1)(kL)B1(m2)exp[im2(ϕπ/2)].
E3sc(r2,ϕ2)=mexp(βm)B1(m)m1Hm+m1(1)(kL)B2(m1)×m2Hm1m2(1)(kL)B1(m2)×m3Hm2+m3(1)(kL)Jm3(kr2)exp(im3ϕ2).
E4sc(r,ϕ)(2πkr)1/2exp[i(krπ/4+kLcosϕ)]mexp(βm)B1(m)m1Hm+m1(1)(kL)B2(m1)m2Hm1m2(1)(kL)B1(m2)×m3Hm2+m3(1)(kL)B2(m3)exp[im3(ϕπ/2)].
Esc(r,ϕ)=E1sc+E2sc+E3sc+E4sc+(2πkr)1/2exp[i(krπ/4)][mexp(βm)B1(m)exp[im(ϕπ/2)]+m,m1,m2exp(βm2)B1(m2)Hm2+m1(1)(kL)B2(m1)Hm1m(1)(kL)B1(m)exp[im(ϕπ/2)]++exp(ikLcosϕ){m,m1exp(βm1)B1(m1)Hm1+m(1)(kL)B2(m)exp[im(ϕπ/2)]+m,m1,m2,m3exp(βm3)B1(m3)Hm3+m1(1)(kL)B2(m1)Hm1m2(1)(kL)B1(m2)Hm2+m(1)(kL)B2(m)exp[im(ϕπ/2)]+}]=(2πkr)1/2exp[i(krπ/4)]×m{Escat1(m)B1(m)+exp(ikLcosϕ)Escat2(m)B2(m)}exp[im(ϕπ/2)].
Escat1(m)=exp(βm)+I=1m1,m2,m2Iexp(βm2I)×i=1IB1(m2i)Hm2i+m2i1(1)(kL)B2(m2i1)Hm2i1m2i2(1)(kL),
Escat2(m)=m1B1(m1)Hm1+m(1)(kL)×{exp(βm1)+I=1m1,m2,m2I+1exp(βm2I+1)×i=1IB1(m2i+1)Hm2i+1+m2i(1)(kL)B2(m2i)Hm2im2i1(1)(kL)}.
σ=kr|Esc|2=2π|m{Escat1(m)B1(m)+exp(ikLcosϕ)Escat2(m)B2(m)}exp[im(ϕπ/2)]|2.
UI(n1,D1,L,ϕin)={i[IiI0σ(ϕi;n1,D1,L,ϕin)]2N}1/2/(iIiN).
|EWPW|2=(1+{n2cos(π4ϕPin)/[1n22sin2(π4ϕPin)1/2]}1+{n2cos(π4ϕEin)/[1n22sin2(π4ϕEin)1/2]})2{1+[(1n22sin2ϕPin)1/2/(n2cosϕPin)]}2(n221)/(n22cos2ϕEin)exp[κ(2h+D1)],
|EWPW|2=[1+(cos(π4ϕPin)/{n2[1n22sin2(π4ϕPin)1/2]})1+(cos(π4ϕEin)/{n2[1n22sin2(π4ϕEin)1/2]})]2{1+[n2(1n22sin2ϕPin)1/2/cosϕPin]}21+[n22(n22sin2ϕEin1)/cos2ϕEin]exp[κ(2h+D1)].

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