Abstract

Reflectance (R), transmittance (T), and absorptance (A) are calculated for a thin-film stack illuminated by a focused field. Based on Debye’s integral representation, the electric and magnetic fields near focus are obtained, and the formulas for R, T, and A are represented as integrals of Poynting vectors. This formulation is applied to the case of a numerical aperture (N.A.) greater than 1.0 as well as to the case of a N.A. less than 1.0, and the corresponding numerical results are presented. They reveal that R, T, and A vary with N.A. and that the amount of variation increases with layer thickness.

© 2000 Optical Society of America

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References

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  1. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
    [CrossRef]
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical system. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  3. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  4. T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
    [CrossRef]
  5. T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
    [CrossRef]
  6. J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–225 (1996).
    [CrossRef]
  7. V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
    [CrossRef]
  8. H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  9. P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  10. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  11. P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  12. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  13. D. Jiang, J. J. Stamnes, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
    [CrossRef]
  14. D. Jiang, J. J. Stamnes, “Theoretical and experimental results for 2D electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
    [CrossRef]
  15. D. G. Flagello, T. Milster, “3d modeling of high numerical aperture imaging in thin films,” in Design, Modeling and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. SPIE1625, 246–261 (1992).
    [CrossRef]
  16. D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
    [CrossRef]
  17. K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
    [CrossRef]
  18. S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
    [CrossRef]
  19. B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
    [CrossRef]
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Sec. 13.1.
  21. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Sec. 9.7.
  22. H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, New York, 1986), Chap. 2.
  23. M. Mansuripur, G. A. Neville Connell, J. W. Goodman, “Laser-induced local heating of multilayers,” Appl. Opt. 21, 1106–1114 (1982).
    [CrossRef] [PubMed]
  24. A. H. A. Holtslag, “Calculations on temperature profiles in optical recording,” J. Appl. Phys. 66, 1530–1543 (1989).
    [CrossRef]

1998 (3)

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for 2D electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

1997 (2)

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

1996 (2)

1995 (3)

1994 (2)

W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

1989 (1)

A. H. A. Holtslag, “Calculations on temperature profiles in optical recording,” J. Appl. Phys. 66, 1530–1543 (1989).
[CrossRef]

1984 (1)

1982 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical system. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Barakat, R.

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Sec. 13.1.

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–225 (1996).
[CrossRef]

Flagello, D. G.

D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
[CrossRef]

D. G. Flagello, T. Milster, “3d modeling of high numerical aperture imaging in thin films,” in Design, Modeling and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. SPIE1625, 246–261 (1992).
[CrossRef]

Goodman, J. W.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Sec. 9.7.

Holtslag, A. H. A.

A. H. A. Holtslag, “Calculations on temperature profiles in optical recording,” J. Appl. Phys. 66, 1530–1543 (1989).
[CrossRef]

Hsu, W.

Ichimura, I.

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

Jiang, D.

D. Jiang, J. J. Stamnes, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for 2D electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Kino, G. S.

Laczik, Z.

Lee, S.-W.

Ling, H.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, New York, 1986), Chap. 2.

Maeda, F.

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

Mamin, H. J.

B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Mansfield, S. M.

Mansuripur, M.

Milster, T.

D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
[CrossRef]

D. G. Flagello, T. Milster, “3d modeling of high numerical aperture imaging in thin films,” in Design, Modeling and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. SPIE1625, 246–261 (1992).
[CrossRef]

Németh, G.

Neville Connell, G. A.

Osato, K.

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

S. M. Mansfield, W. R. Studenmund, G. S. Kino, K. Osato, “High-numerical-aperture lens system for optical storage,” Opt. Lett. 18, 305–307 (1993).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical system. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Rosenbluth, A. E.

Rugar, D.

B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Stamnes, J. J.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for 2D electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–225 (1996).
[CrossRef]

Studenmund, W. R.

Terris, B. D.

B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

Török, P.

Varga, P.

Visser, T. D.

Watanabe, T.

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

Wiersma, S. H.

Wolf, E.

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical system. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Sec. 13.1.

Yamamoto, K.

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

B. D. Terris, H. J. Mamin, D. Rugar, “Near-field optical data storage using a solid immersion lens,” Appl. Phys. Lett. 65, 388–390 (1994).
[CrossRef]

J. Appl. Phys. (1)

A. H. A. Holtslag, “Calculations on temperature profiles in optical recording,” J. Appl. Phys. 66, 1530–1543 (1989).
[CrossRef]

J. Opt. Soc. Am. A (8)

H. Ling, S.-W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

P. Török, P. Varga, Z. Laczik, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
[CrossRef]

T. D. Visser, S. H. Wiersma, “Spherical aberration and the electromagnetic field in high aperture systems,” J. Opt. Soc. Am. A 8, 1404–1410 (1991).
[CrossRef]

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[CrossRef]

D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
[CrossRef]

Jpn. J. Appl. Phys. Part 1 (1)

K. Yamamoto, K. Osato, I. Ichimura, F. Maeda, T. Watanabe, “0.8 Numerical aperture two-element objective lens for the optical disk,” Jpn. J. Appl. Phys. Part 1 36, 456–459 (1997).
[CrossRef]

Opt. Lett. (1)

Proc. R. Soc. London, Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical system. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt. (5)

J. J. Stamnes, V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–225 (1996).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for 2D electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

Other (4)

D. G. Flagello, T. Milster, “3d modeling of high numerical aperture imaging in thin films,” in Design, Modeling and Control of Laser Beam Optics, Y. Kohanzadeh, G. N. Laurence, J. G. McCoy, H. Weichel, eds., Proc. SPIE1625, 246–261 (1992).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Sec. 13.1.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Sec. 9.7.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, New York, 1986), Chap. 2.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a thin-film stack illuminated by a focused field. The solid angle Ω is subtended by the exit pupil, and the thin-film stack is located near the focus.

Fig. 2
Fig. 2

Thin-film stack composed of two 50-nm Si3N4 layers and a 100-nm a-Si layer between them.

Fig. 3
Fig. 3

Magnitude of the Poynting vector along the propagating direction for N.A. of 0.0, 0.5, and 0.85. The N.A. of 0.0 corresponds to the case of a normally incident plane wave.

Fig. 4
Fig. 4

R, T, and A of the thin-film stack depicted in Fig. 2 versus N.A. in the range 0.0–1.0.

Fig. 5
Fig. 5

Reflectivity of the s- and p-polarized waves versus incident angle of a plane wave for the thin-film stack depicted in Fig. 2.

Fig. 6
Fig. 6

Reflectivity versus the thickness of the upper Si3N4 layer in the thin-film stack depicted in Fig. 2 for N.A. of 0.0, 0.6, and 0.85.

Fig. 7
Fig. 7

Schematic diagram of a SIL and a thin-film stack. The film structure is identical to that of Fig. 2, and the refractive index of the SIL is 1.9.

Fig. 8
Fig. 8

R, T, and A versus N.A. for the system of Fig. 7. The N.A. range is 0–1.5.

Fig. 9
Fig. 9

Reflectivity of the s- and p-polarized waves versus incident angle of the plane wave for a thin-film stack that is composed of the incident medium, air gap, and thin-film stack of Fig. 2.

Fig. 10
Fig. 10

Distribution of the energy-flow rate in the a-Si layer of Fig. 2, imaged by lenses with N.A. of 0.6, 0.85, and 1.2.

Equations (54)

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

E(r)=-ik2πΩE(θ0, ϕ0, r)dΩ,
E(θ0, ϕ0, r)=E0cos θ0P(θ0, ϕ0)exp(ik0r).
Pxs=sin2 ϕ0,
Pxp=cos2 ϕ0cos θ0,
Pys=-sin ϕ0cos θ0,
Pyp=cos ϕ0sin ϕ0cos θ0,
Pzp=-cos ϕ0sin θ0.
k0r=k0n0(α0x+β0y+γ0z),
α0=cos ϕ0sin θ0,
β0=sin ϕ0sin θ0,
γ0=cos θ0.
E(x, y, z)=Ej+1,i(θ, ϕ)expikj(αjx+βjy)-ikjγjl=1jdl-z+Ej+1,r(θ, ϕ)expikj(αjx+βjy)+ikjγjl=1jdl-z,
njαj=n0α0,
njβj=n0β0,
njγj=(nj2-n02sin2 θ0)1/2.
Ej+1,i=Ej+1,is+Ej+1, jp,
Ej+1,r=Ej+1,rs+Ej+1,rp.
tj+1s=EN+1,t,xsEj+1,i,xs=EN+1,t,ysEj+1,i,ys,
rj+1s=Ej+1,r,xsEj+1,i,xs=Ej+1,r,ysEj+1,i,ys,
Ej+1,i,xs=t1stj+1s E1,i,xs,
Ej+1,r,xs=t1stj+1s rj+1sE1,i,xs.
E=0or kE=0,
EN+1,t,xpEj+1,i,xp=EN+1,tpcos θN+1,tEj+1,ipcos θj+1,i=tj+1pcos θN+1,tcos θj+1,i,
EN+1,r,xpEj+1,i,xp=-EN+1,rpcos θj+1,rEj+1,ipcos θj+1,i=-rj+1p.
Ej+1,i,xp=t1ptj+1pcos θj+1,icos θ1,i E1,i,xp,
Ej+1,r,xp=-rj+1pt1ptj+1pcos θj+1,icos θ1,i E1,i,xp=-rj+1pt1ptj+1p[1-(n0/nj)2sin2 θ0]1/2cos θ0.
EN+1,t,zpEj+1,i,zp=(-EN+1,tpsin θN+1,t)(-Ej+1,ipsin θj+1,i)=t1pnjnN+1,
Ej+1,r,zpEj+1,i,zp=(-Ej+1,rpsin θj+1,r)(-Ej+1,ipsin θj+1,i)=rj+1p.
Ej+1,i,zp=t1ptj+1pn0nj E1,i,zp,
Ej+1,r,zp=rj+1pt1ptj+1pn0nj E1,i,zp.
Ex(θj, ϕj, r)=E1,i,xs(θ0, ϕ0)exp[ik0(α0x+β0y)] t1stj+1s×[exp(-iΦ)+rj+1sexp(iΦ)]+E1,i,xp(θ0, ϕ0)exp[ik0(α0x+β0y)]×t1ptj+1p[1-(n0/nj)2sin2 θ0]1/2cos θ0×[exp(-iΦ)-rj+1pexp(iΦ)],
Ez(θj, ϕj, r)=E1,i,zp(θ0, ϕ0)exp[ik0(α0x+β0y)]×t1ptj+1pn0nj [exp(-iΦ)+rj+1pexp(iΦ)],
Φ=kjγjl=1jdl-z.
E1,i,xs=E0(θ0, ϕ0)cos θ0Pxs(θ0, ϕ0)exp(-ikzΔz),
E1,i,xp=E0(θ0, ϕ0)cos θ0Pxp(θ0, ϕ0)exp(-ikzΔz)
Ex(r)=-ik2πΩ E0(θ0, ϕ0)cos θ0Pxs(θ0, ϕ0)×exp[ik0(α0x+β0y-γ0Δz)]×t1stj+1s [exp(-iΦ)+rj+1sexp(iΦ)]dΩ- ik2πΩ E0(θ0, ϕ0)cos θ0Pxp(θ0, ϕ0)×exp[ik0(α0x+β0y-γ0Δz)]×t1ptj+1p[1-(n0/nj)2sin2 θ0]1/2cos θ0×[exp(-iΦ)-rj+1pexp(iΦ)]dΩ.
Ey(r)=-ik2πΩE0(θ0, ϕ0)cos θ0Pys(θ0, ϕ0)×exp[ik0(α0x+β0y-γ0Δz)]×t1stj+1s [exp(-iΦ)+rj+1sexp(iΦ)]dΩ-ik2πΩE0(θ0, ϕ0)cos θ0Pyp(θ0, ϕ0)×exp[ik0(α0x+β0y-γ0Δz)]×t1ptj+1p[1-(n0/nj)2sin2 θ0]1/2cos θ0×[exp(-iΦ)-rj+1pexp(iΦ)]dΩ0,
Ez(r)=-ik2πΩE0(θ0, ϕ0)cos θ0Pzp(θ0, ϕ0)×exp[ik0(α0x+β0y-γ0Δz)]×t1ptj+1pn0nj [exp(-iΦ)+rj+1p×exp(iΦ)]dΩ0.
rj+1s=Yjm11+YjYN+1m12-m21-YN+1m22Yjm11+YjYN+1m12+m21+YN+1m22,
tj+1s=2YjYjm11+YjYN+1m12+m21+YN+1m22
rj+1p=-Yjm11-YjYN+1m12+m21+YN+1m22Yjm11+YjYN+1m12+m21+YN+1m22,
tj+1p=njnv+12YN+1Yjm11+YjYN+1m12+m21+YN+1m22
m11m12m21m22=M˜=M˜j+1M˜j+2M˜N.
M˜j+1=cos(k0hj+1)-i sin(k0hj+1)Yj+1-Yj+1i sin(k0hj+1)cos(k0 hj+1),
hj+1=nj+1dj+1cos θj+1,
Yj+1=ε0/μ0nj+1cos θj+1,s-polarizedwave,
Yj+1=ε0/μ0nj+1cos θj+1 p-polarizedwave.
S=c4πRe[E×H*],
H(r)=-ik2πεμΩk(θj, ϕj)×E(θj, ϕj, r)dΩ.
I(z)= Sz(x, y, z)dxdy.
R=1-I(zj)I0,
T=I(zN+1)I0,
A=I(zj+1)-I(zj)I0,
g(x, y, z)=-ddz Sz(x, y, z).

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