Abstract

We have developed a numerical diffraction tool for cases in which the incident field is a focused spot and the diffracting structure is a single structure or an aperiodic surface. Our approach uses the integral formulation to solve Maxwell's equations and is different from previously published methods in its choice of basis function. We compared numerical results with experimental measurements of the far-field intensity for a focused spot incident on an aluminum grating, and the comparison was favorable. Finally, we predict the diffraction behavior of the proposed digital video disk format for the next generation of optical disk. Our analysis shows that the reflected signal for this format has a strong dependence on the polarization of the incident light.

© 1997 Optical Society of America

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References

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  1. J. G. Dil, B. A. Jacobs, “Apparent size of reflecting polygonal obstacles of the order of one wavelength,” J. Opt. Soc. Am. 69, 950–960 (1979).
    [CrossRef]
  2. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.
  3. D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
    [CrossRef]
  4. A. K. Fung, M. F. Chen, “Numerical simulation of scattering from simple and composite random surfaces,” J. Opt. Soc. Am. A 2, 2274–2284 (1985).
    [CrossRef]
  5. G. S. White, J. F. Marchiando, “Scattering from a V-shaped groove in the resonance domain,” Appl. Opt. 22, 2308–2312 (1983).
    [CrossRef] [PubMed]
  6. E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
    [CrossRef]
  7. The National Technology Roadmap for Semiconductors (Semiconductor Industry Association, San Jose, Calif., 1994).
  8. C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
    [CrossRef]
  9. C. J. Raymond, S. S. H. Naqvi, J. R. McNeil, “Scatterometry for CD measurements of etched structures,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 720–728 (1996).
    [CrossRef]
  10. R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  11. M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  12. J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
    [CrossRef]
  13. A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
    [CrossRef]
  14. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  15. F. G. Kaspar, “Diffraction by thick, periodically modified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  16. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  17. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  18. T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [CrossRef]
  19. W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).
  20. A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conf. Publ. 13 (Peter Peregrinus Ltd., Stevenage, UK, 1977), Chap. 6.
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  22. L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, 1985).
  23. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives: erratum,” J. Opt. Soc. Am. A 10, 382–383 (1993).
    [CrossRef]
  24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 6.
  25. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.
  26. E. Wolf, “Electromagnetic diffraction in optical systems. 1. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
    [CrossRef]
  27. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  28. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  29. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

1993 (1)

1987 (1)

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

1985 (1)

1984 (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

1983 (1)

1982 (2)

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

1981 (1)

1979 (1)

1978 (2)

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[CrossRef]

1977 (1)

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

1973 (2)

M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

F. G. Kaspar, “Diffraction by thick, periodically modified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
[CrossRef]

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems. 1. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Bouwhuis, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Braat, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Cadilhac, M.

M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Chen, C.-L.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

Chen, M. F.

De Hoop, A. T.

A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conf. Publ. 13 (Peter Peregrinus Ltd., Stevenage, UK, 1977), Chap. 6.

Delves, L. M.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, 1985).

Dil, J. G.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Fung, A. K.

Gaither, S. A.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

Gaylord, T. K.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Hosch, J. W.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

Hugonin, J. P.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Huijser, A.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Immink, K. S.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Jacobs, B. A.

Kaspar, F. G.

Knop, K.

Mansuripur, M.

Marchiando, J. F.

Maystre, D.

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

McNeil, J. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

C. J. Raymond, S. S. H. Naqvi, J. R. McNeil, “Scatterometry for CD measurements of etched structures,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 720–728 (1996).
[CrossRef]

Mendez, E. R.

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Mohamed, J. L.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, 1985).

Moharam, M. G.

Murnane, M. R.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

Naqvi, S. S. H.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

C. J. Raymond, S. S. H. Naqvi, J. R. McNeil, “Scatterometry for CD measurements of etched structures,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 720–728 (1996).
[CrossRef]

Neviere, M.

M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

O’Donnell, K. A.

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

Osborne, T. R.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

Pasman, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Petit, R.

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Prins, S. L.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

Raymond, C. J.

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

C. J. Raymond, S. S. H. Naqvi, J. R. McNeil, “Scatterometry for CD measurements of etched structures,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 720–728 (1996).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 6.

Swanson, G. J.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

van Rosmalen, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

Veldkamp, W. B.

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

White, G. S.

Wirgin, A.

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[CrossRef]

Wolf, E.

E. Wolf, “Electromagnetic diffraction in optical systems. 1. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

Appl. Opt. (1)

Appl. Phys. B (1)

T. K. Gaylord, M. G. Moharam, “Planar dielectric grating diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

J. Opt. (Paris) (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

J. Opt. Soc. Am. (5)

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

Opt. Commun. (4)

E. R. Mendez, K. A. O’Donnell, “Observation of depolarization and backscattering enhancement in light scattering from Gaussian random surfaces,” Opt. Commun. 61, 91–95 (1987).
[CrossRef]

M. Neviere, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

J. P. Hugonin, R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

E. Wolf, “Electromagnetic diffraction in optical systems. 1. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959).
[CrossRef]

Other (14)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

W. B. Veldkamp, G. J. Swanson, S. A. Gaither, C.-L. Chen, T. R. Osborne, “Binary optics: a diffraction analysis,” (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass.1989).

A. T. De Hoop, Modern Topics in Electromagnetics and Antennas, PPL Conf. Publ. 13 (Peter Peregrinus Ltd., Stevenage, UK, 1977), Chap. 6.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, 1985).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 6.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, K. S. Immink, Principles of Optical Disc Systems (Hilger, Bristol, UK, 1985), Chap. 6.

The National Technology Roadmap for Semiconductors (Semiconductor Industry Association, San Jose, Calif., 1994).

C. J. Raymond, M. R. Murnane, S. L. Prins, S. S. H. Naqvi, J. R. McNeil, J. W. Hosch, “Multi-parameter CD measurements using scatterometry,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 698–709 (1996).
[CrossRef]

C. J. Raymond, S. S. H. Naqvi, J. R. McNeil, “Scatterometry for CD measurements of etched structures,” in Metrology Inspection and Process Control for Microlithography X, S. K. Jones, ed., Proc. SPIE2725, 720–728 (1996).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Basic diffraction model addressed in this paper. NA stands for numerical aperture.  

Fig. 2
Fig. 2

Diffracting structure and coordinate systems for a two-dimensional problem.

Fig. 3
Fig. 3

Coordinate system for the far-field calculation.

Fig. 4
Fig. 4

Setup for measurement of diffracted far field for focused incident fields.

Fig. 5
Fig. 5

Reflected Fourier plane for aluminum grating with TM incidence, clearly showing the Wood's anomalies.

Fig. 6
Fig. 6

Measurement versus calculation for the sinusoidal aluminum grating with the Wood's anomalies evident as minima in the angular spectrum of the reflected field. The aperture limitations to the incident and reflected fields are indicated by dashed vertical lines.

Fig. 7
Fig. 7

Calculated pit phase depth and far-field intensity for the DVD format. The dashed vertical lines indicate which pit depths can be used to represent four logical states.

Fig. 8
Fig. 8

Pit phase depth for different pit widths with the pit depth equal to a quarter wave. In the region where the pit width is between λ/2 (λ is adjusted for the index of refraction of the incident medium) and λ, the phase depth for TE illumination varies almost a quarter wave, but it remains fairly constant for TM illumination.

Equations (42)

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

J(rs)=Js, m(rs)Js, e(rs)=nˆ×E(rs)-nˆ×H(rs),
J(r)=ΓJ(rs),
Γ=-nˆ××[G(r; rs) * nˆ×jωμ(1+2/k2)[G(r; rs) * nˆ×jω(1+2/k2)[G(r; rs) * nˆ××[G(r; rs) * .
G(r; r)=exp(jk|r-r|)4π|r-r|,
G(r; rs)*J(rs)= G(r; rs)J(rs)drs.
[Γ(1)-Γ(0)-1]J=-J(i).
G(r; rs)=- exp(jk|r-rs|)4π|r-rs| dy=j4 H0(1)(k|r-rs|),
Γ11Js, m(rs)=-nˆ(s)××-[G1(R)-G0(R)]×Js, m(s)τˆ(s)ds,
Γ12Js, e(rs)=jωμnˆ(s)×-[G1(R)-G0(R)]×Js, e(s)yˆ ds,
Γ21Js, m(rs)=nˆ(s)jωμ×-[k12G1(R)-k02G0(R)]×Js, m(s)τˆ(s)ds+2-[G1(R)-G0(R)]×Js, m(s)τˆ(s)ds,
Γ22Js, e(rs)=-nˆ(s)××-[G1(R)-G0(R)]×Js, e(s)yˆ ds,
Js, m(s)=l=-NN al sinc(k0Ws-πl),
Js, e(s)=l=-NN bl sinc(k0Ws-πl).
Γ11l=-nˆ(s)××-[G1(R)-G0(R)]×sinc(k0Ws-πl)τˆ(s)ds,
Γ12l=jωμτˆ(s)-[G1(R)-G0(R)]×sinc(k0Ws-πl)ds,
Γ21l=nˆ(s)jωμ×-[k12G1(R)-k02G0(R)]×sinc(k0Ws-πl)ds+2-[G1(R)-G0(R)]×sinc(k0Ws-πl)dsτˆ(s),
Γ22l=yˆ[nˆ(s)]- [G1(R)-G0(R)]×sinc(k0Ws-πl)ds,
E(i)(x, z)=yˆB(u)exp[jk0(ux+1-u2z)]du.
Hx(i)(x, z)=B(u)1-u2×exp[jk0(ux+1-u2z)]du,
Hz(i)(x, z)=B(u)×uexp[jk0(ux+1-u2z)]du.
B(u)=1,0,|u|NA|u|>NA,
B(u)=1/1-u2,0,|u|NA|u|>NA,
Js, m(i)(s)=nˆ×E(i)=τˆ(s)Ey(i)(xs, zs),
Js, e(i)(s)=-nˆ×H(i)=yˆ[nx(s)Hz(i)(xs, zs)-nz(s)Hx(i)(xs, zs)].
-E(θ)=yˆ2k0π expj π4 ×-{[nx(s)sin θ-nz(s)cos θ]×Js, m(s)+Js, e(s)}×exp{-jk0[x(s)sin θ-z(s)cos θ]}ds.
P=- Re(E×H*)nˆ ds.
Js, m×Js, e*=(-nˆ)[nˆ(E×H*)].
Js, mτˆ×Js, e*yˆ=-Js, mJs, e*nˆ,
P=- Re[Js, m(s)Js, e*(s)]ds.
P=π2W l Re(albl*).
- sinc(k0Ws-πl)sinc(k0Ws-πl)ds
=π2k0W δll,
P=P(i)-P(r),
Js, m(s)=Js, m(i)(s)+Js, m(r)(s),
Js, e(s)=Js, e(i)(s)+Js, e(r)(s).
al(r)=al-al(i),bl(r)=bl-bl(i).
Ey=sin(pπx/d)exp(jkzz),kz=[k2-(pπ/d)2]1/2,
Hy=cos(pπx/d)exp(jkzz),kz=[k2-(pπ/d)2]1/2.
Js, e=Js, eτˆ(s),Js, m=-Js, myˆ.
Γ11lTM=Γ22lTE,Γ22lTM=Γ11lTE.
Γ12TMJs, e(rs)=jωμnˆ(s)×-[G1(R)-G0(R)]×Js, e(s)τˆ(s)ds+2-G1(R)k12-G0(R)k02×Js, e(s)τˆ(s)ds,
Γ21TMJs, m(rs)=-nˆ(s)jωμ×-[k12G1(R)-k02G0(R)]×Js, m(s)yˆ ds.

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