Abstract

Reflectivity of a random monolayer, consisting of transparent spherical particles, is estimated using a graded refractive index model, an effective medium approach, and two scattering models. Two cases, a self-standing film and one with a substrate, are considered. Neither the surrounding medium nor the substrate are absorbing materials. Results at normal incidence, with different particle sizes, covering ratios and refractive indexes, are compared. The purpose of this work is to find under which circumstances, for reflectivity at normal incidence, a particle monolayer behaves as a graded refractive index film.

© 2012 Optical Society of America

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References

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  1. T. Yamaguchi, H. Takahashi, and A. Sudoh, “Optical behavior of a metal island film,” J. Opt. Soc. Am. 68, 1039–1044 (1978).
    [CrossRef]
  2. R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
    [CrossRef]
  3. R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
    [CrossRef]
  4. T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
    [CrossRef]
  5. T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor with gold colloid monolayers deposited upon glass substrates,” Opt. Lett. 25, 372–374 (2000).
    [CrossRef]
  6. J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
    [CrossRef]
  7. R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
    [CrossRef]
  8. A. García-Valenzuela, E. Gutiérrez-Reyes, and R. G. Barrera, “Multiple-scattering model for the coherent reflection and transmission of light from a disordered monolayer of particles,” J. Opt. Soc. Am. A 29, 1161–1179 (2012).
    [CrossRef]
  9. R. Diamant and M. Fernández-Guasti, “Light propagation in 1D inhomogeneous deterministic media: the effect of discontinuities,” J. Opt. A 11, 045712 (2009).
    [CrossRef]
  10. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1981).
  11. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  12. M. Mishchenko, Light Scattering by Nonspherical Particles. Theory, Measurements, and Applications (Academic, 1999).
  13. V. Bazhan, “ScatLab,” http://www.scatlab.org .
  14. V. A. Loiko, V. P. Dick, and V. I. Molochko, “Monolayers of discrete scatterers: comparison of the single-scattering and quasi-crystalline approximations,” J. Opt. Soc. Am. A 15, 2351–2354 (1998).
    [CrossRef]
  15. M. C. Peña-Gomar, J. J. F. Castillo, A. García-Valenzuela, R. G. Barrera, and E. Pérez, “Coherent optical reflectance from a monolayer of large particles adsorbed on a glass surface,” Appl. Opt. 45, 626–632 (2006).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Cambridge University, 2005).
  17. E. Hecht Eugene, Optics (Addison Wesley, 2002).
  18. M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).
  19. J. L. Reid and J. R. Ray, “Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion,” J. Math. Phys. 21, 1583–1587 (1980).
    [CrossRef]
  20. P. B. Espinoza-Padilla, “Ermakov-Lewis dynamic invariants with some applications,” http://arxiv.org/pdf/math-ph/0002005 .
  21. N. Atzin, M. Fernandez–Guasti, and R. Diamant, “Light propagation at soft interface,” http://demonstrations.wolfram.com/LightPropagationAtSoftInterface/ .
  22. P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. USA 16, 627–637 (1930).
    [CrossRef]
  23. A. M. Goodman, “Optical interference method for the approximate determination of refractive index and thickness of a transparent layer,” Appl. Opt. 17, 2779–2787 (1978).
    [CrossRef]
  24. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, Vol. V, E. Wolf, ed. (North-Holland, 1966), pp. 248–286.

2012 (1)

2009 (2)

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

R. Diamant and M. Fernández-Guasti, “Light propagation in 1D inhomogeneous deterministic media: the effect of discontinuities,” J. Opt. A 11, 045712 (2009).
[CrossRef]

2008 (1)

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

2006 (1)

2000 (2)

T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor with gold colloid monolayers deposited upon glass substrates,” Opt. Lett. 25, 372–374 (2000).
[CrossRef]

M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).

1999 (1)

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

1998 (2)

V. A. Loiko, V. P. Dick, and V. I. Molochko, “Monolayers of discrete scatterers: comparison of the single-scattering and quasi-crystalline approximations,” J. Opt. Soc. Am. A 15, 2351–2354 (1998).
[CrossRef]

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

1991 (1)

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

1980 (1)

J. L. Reid and J. R. Ray, “Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion,” J. Math. Phys. 21, 1583–1587 (1980).
[CrossRef]

1978 (2)

1930 (1)

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. USA 16, 627–637 (1930).
[CrossRef]

Afonso, C. N.

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

Babonneau, D.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Ballesteros, J. M.

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

Barrera, R. G.

Bohren, C. F.

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

Borensztein, Y.

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2005).

Bosbach, J.

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

Camelio, S.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Castillo, J. J. F.

Castillo-Mussot, M. del

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

de Sande, J. C. G.

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

Diamant, R.

R. Diamant and M. Fernández-Guasti, “Light propagation in 1D inhomogeneous deterministic media: the effect of discontinuities,” J. Opt. A 11, 045712 (2009).
[CrossRef]

M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).

Dick, V. P.

Epstein, P. S.

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. USA 16, 627–637 (1930).
[CrossRef]

Eugene, E. Hecht

E. Hecht Eugene, Optics (Addison Wesley, 2002).

Fernández-Guasti, M.

R. Diamant and M. Fernández-Guasti, “Light propagation in 1D inhomogeneous deterministic media: the effect of discontinuities,” J. Opt. A 11, 045712 (2009).
[CrossRef]

M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).

García-Valenzuela, A.

Gil-Villegas, A.

M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).

Girardeau, T.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Goodman, A. M.

Gutiérrez-Reyes, E.

Huffman, D. R.

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

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, Vol. V, E. Wolf, ed. (North-Holland, 1966), pp. 248–286.

Jupille, J.

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

Kobayashi, T.

Lazzari, R.

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

Loiko, V. A.

Mishchenko, M.

M. Mishchenko, Light Scattering by Nonspherical Particles. Theory, Measurements, and Applications (Academic, 1999).

Mochán, W. L.

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

Molochko, V. I.

Monsivais, G.

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

Okamoto, T.

Peña-Gomar, M. C.

Pérez, E.

Ray, J. R.

J. L. Reid and J. R. Ray, “Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion,” J. Math. Phys. 21, 1583–1587 (1980).
[CrossRef]

Reid, J. L.

J. L. Reid and J. R. Ray, “Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion,” J. Math. Phys. 21, 1583–1587 (1980).
[CrossRef]

Renaud, G.

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

Revenant, C.

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

Serna, R.

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

Simonot, L.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Stietz, F.

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

Sudoh, A.

Takahashi, H.

Toudert, J.

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Träger, F.

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1981).

Villaseñor, P.

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

Wenzel, T.

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2005).

Yamaguchi, I.

Yamaguchi, T.

Appl. Opt. (2)

J. Appl. Phys. (1)

R. Serna, J. C. G. de Sande, J. M. Ballesteros, and C. N. Afonso, “Spectroscopic ellipsometry of composite thin films with embedded Bi nanocrystals,” J. Appl. Phys. 84, 4509–4516 (1998).
[CrossRef]

J. Math. Phys. (1)

J. L. Reid and J. R. Ray, “Ermakov systems, nonlinear superposition and solutions of nonlinear equations of motion,” J. Math. Phys. 21, 1583–1587 (1980).
[CrossRef]

J. Opt. A (1)

R. Diamant and M. Fernández-Guasti, “Light propagation in 1D inhomogeneous deterministic media: the effect of discontinuities,” J. Opt. A 11, 045712 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nanotechnology (1)

J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers: the influence of nanocluster size, shape and organization,” Nanotechnology 19, 125709 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

R. Lazzari, G. Renaud, C. Revenant, J. Jupille, and Y. Borensztein, “Adhesion of growing nanoparticles at a glance: surface differential reflectivity spectroscopy and grazing incidence small angle x-ray scattering,” Phys. Rev. B 79, 125428 (2009).
[CrossRef]

R. G. Barrera, M. del Castillo-Mussot, G. Monsivais, P. Villaseñor, and W. L. Mochán, “Optical properties of 2D disordered systems on a substrate,” Phys. Rev. B 43, 13819–13826 (1991).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

P. S. Epstein, “Reflection of waves in an inhomogeneous absorbing medium,” Proc. Natl. Acad. Sci. USA 16, 627–637 (1930).
[CrossRef]

Rev. Mex. Fís. (1)

M. Fernández-Guasti, A. Gil-Villegas, and R. Diamant, “Ermakov Equation Arising from Electromagnetic Fields Propagating in 1D Inhomogeneous Media,” Rev. Mex. Fís. 46, 530–538 (2000).

Surf. Sci. (1)

T. Wenzel, J. Bosbach, F. Stietz, and F. Träger, “In situ determination of the shape of supported silver clusters during growth,” Surf. Sci. 432, 257–264 (1999).
[CrossRef]

Other (9)

P. B. Espinoza-Padilla, “Ermakov-Lewis dynamic invariants with some applications,” http://arxiv.org/pdf/math-ph/0002005 .

N. Atzin, M. Fernandez–Guasti, and R. Diamant, “Light propagation at soft interface,” http://demonstrations.wolfram.com/LightPropagationAtSoftInterface/ .

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, 1981).

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

M. Mishchenko, Light Scattering by Nonspherical Particles. Theory, Measurements, and Applications (Academic, 1999).

V. Bazhan, “ScatLab,” http://www.scatlab.org .

M. Born and E. Wolf, Principles of Optics (Cambridge University, 2005).

E. Hecht Eugene, Optics (Addison Wesley, 2002).

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, Vol. V, E. Wolf, ed. (North-Holland, 1966), pp. 248–286.

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Figures (8)

Fig. 1.
Fig. 1.

Monolayer in both cases, (a) without and (b) with a substrate. Refractive indexes n1, n2, and n3 belong to the surrounding medium, spheres, and substrate, respectively. The curly arrows represent incident, reflected, and transmitted light. Scattered light is not shown.

Fig. 2.
Fig. 2.

Consider one particle and any plane perpendicular to the z axis between z=2a and z=0.

Fig. 3.
Fig. 3.

n(z) in both cases, (a) without and (b) with a substrate.

Fig. 4.
Fig. 4.

Reflectivity versus radius (a) for a monolayer with refractive index n2=1.3 and covering ratio Θ=0.3, (b) for n2=1.1 and Θ=0.1, (c) for n2=1.5 and Θ=0.3. MSM-M is the multiple-scattering model and SSA-M is the single-scattering approximation, both based on the Mie solution. EGL is the equivalent stratified layer and THF is a thin homogeneous film.

Fig. 5.
Fig. 5.

Reflectivity versus radius for a monolayer on a substrate with n3=1.5. MSM-M is multiple-scattering model and SSA-M is single-scattering approximation, both based on the Mie solution. EGL is the equivalent stratified layer and THF is a thin homogeneous film. (a) The particles refractive index is n2=1.3 and covering ratio is Θ=0.3; (b) n2=1.1 and Θ=0.1; (c) n2=1.5 and Θ=0.3.

Fig. 6.
Fig. 6.

Reflectivity versus radius, for a monolayer with refractive index n2=1.3 and covering ratio Θ=0.3. EGL is the equivalent stratified layer and THF is a thin homogeneous film. MSM-R&G is the multiple-scattering model and SSA-R&G is the single-scattering approximation, both in turn based on the Rayleigh–Gans approximation, (a) with the standard size parameter definition for scattering models xmn1k0a in Eq. (2), and (b) * with the new size parameter definition xmnewnavgk0a.

Fig. 7.
Fig. 7.

Reflectivity versus radius; MSM-R&G is the multiple-scattering model and SSA-R&G is the single-scattering approximation, both in turn based on the Rayleigh–Gans approximation. EGL is the equivalent stratified layer and THF is a thin homogeneous film, (a) for a monolayer with refractive index n2=1.1 and covering ratio Θ=0.1, and (b) for a monolayer with n2=1.5 and Θ=0.3.

Fig. 8.
Fig. 8.

Reflectivity versus radius for a monolayer on a substrate with n3=1.5. MSM-R&G is the multiple-scattering model and SSA-R&G is the single-scattering approximation, both in turn based on the Rayleigh–Gans approximation. EGL is the equivalent stratified layer and THF is a thin homogeneous film. (a) The particles refractive index is n2=1.3 and covering ratio is Θ=0.3; (b) with n2=1.1 and Θ=0.1, (c) with n2=1.5 and Θ=0.3.

Tables (1)

Tables Icon

Table 1. Ranges for Radii, Refractive Indexes, and Surface Covering Ratios

Equations (18)

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

(ESES)=eik0(rz)ik0r(S2S3S4S1)(EiEi),
S1=ixm3(n22n12n22+2n12)f(θ),
S2=ixm3(n22n12n22+2n12)f(θ)cosθ,
rcohSSA=αSj(πθi),
tcohSSA=1αSj(0).
rcohMSM=αSj(πθi)1+αSj(0)+α24[Sj2(0)Sj2(π2θi)],
tcohMSM=1α24[Sj2(0)Sj2(π2θi)]1+αSj(0)+α24[Sj2(0)Sj2(π2θi)].
rsupp(θi)=rcoh(θi)+rs(θi)tcoh2(θi)eiβ1rs(θi)rcoh(θi)eiβ,
neff2=1+3f(n221n22+2)1f(n221n22+2),
nefff(n21)+1,
R=1(r12r23+1)2(r12+r23)2(r12r23+1)24r12r23sin2(k0navg2a),
n(z)={n1forz2an1+Θ(n2n1)[1(za+1)2]for2a<z<0n3forz0.
dlnμdzExz=k02n2Ex+(2Exy2+2Exz2).
0=k02n2Ex+2Exz2.
AQ2A3=k02n2A,
A2q=Q.
r=AreflectedAincident=AmaxAminAmax+Amin.
2navg(am+1am)=14.

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