Abstract

The Rayleigh methods used in the theory of reflection by a grating have been investigated numerically. The errors made in the power balance and the integrated-square errors made in the fulfillment of the boundary conditions are considered. Only the integrated-square error is a sufficient check for the convergence of our numerical results. The validity of the so-called Rayleigh hypothesis has been confirmed numerically. The minimization of these integrated-square errors leads to a method of general validity; in this method the results are always convergent, and the errors made in the power balance are of the same order as the integrated-square errors in the boundary conditions.

© 1981 Optical Society of America

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References

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  1. A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
    [CrossRef]
  2. Rayleigh (J. W. Strutt), “On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Phil. Mag. 44, 28–52 (1897).
  3. Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, London, 1896) (Dover, New York, 1945); The Theory of Sound, Vol.  II, pp. 89, 297–311.
  4. Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  5. R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  6. J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section,” Proc. IEE 121, 794–804 (1974).
  7. R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–623 (1975).
    [CrossRef]
  8. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
    [CrossRef]
  9. J. A. DeSanto, “Theoretical methods in ocean acoustics,” in Ocean Acoustics (Springer-Verlag, Berlin, 1979), Chap. 2, p. 54.
  10. P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
    [CrossRef]
  11. P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle,” IEEE Trans. Antennas Propag. AP-27, 577–583 (1979).
    [CrossRef]
  12. P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface,” Radio Sci. 15, 723–732 (1980).
    [CrossRef]
  13. J. P. Hugonin, R. Petit, and M. Cadilhac, “On the use of plane wave expansions to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  14. W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
    [CrossRef]
  15. K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzada, eds. (Union Radio-Scientifique Internationale, Brussels, 1971), pp. 257–270.
  16. H. Ikuno and K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [CrossRef]
  17. J. B. Davies, “A least-squares boundary residual method for the numerical solution of scattering problems,” IEEE Trans. Microwave Theory Tech. MTT-21, 99–104 (1973).
    [CrossRef]
  18. H. A. Kalhor, “Numerical evaluation of Rayleigh hypothesis for analyzing scattering from corrugated gratings,” IEEE Trans. Antennas Propag. AP-24, 884–889 (1976).
    [CrossRef]
  19. P. M. van den Berg, “Review of some computational techniques in scattering and diffraction,” in Proceedings International U.R.S.I.-Symposium 1980 (Union Radio-Scientifique Internationale, Brussels, 1980), pp. 211B/1–211B/6.
  20. P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).
  21. R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B. 262, 468–471 (1966).
  22. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
    [CrossRef]
  23. R. Petit and D. Maystre, “Application des lois de l’électromagnétisme, a l’étude des réseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
    [CrossRef]
  24. H. A. Kalhor and A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
    [CrossRef]
  25. K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
    [CrossRef]
  26. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
    [CrossRef]

1981 (1)

1980 (2)

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface,” Radio Sci. 15, 723–732 (1980).
[CrossRef]

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[CrossRef]

1979 (2)

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle,” IEEE Trans. Antennas Propag. AP-27, 577–583 (1979).
[CrossRef]

1976 (1)

H. A. Kalhor, “Numerical evaluation of Rayleigh hypothesis for analyzing scattering from corrugated gratings,” IEEE Trans. Antennas Propag. AP-24, 884–889 (1976).
[CrossRef]

1975 (2)

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–623 (1975).
[CrossRef]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

1974 (1)

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section,” Proc. IEE 121, 794–804 (1974).

1973 (3)

H. Ikuno and K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

J. B. Davies, “A least-squares boundary residual method for the numerical solution of scattering problems,” IEEE Trans. Microwave Theory Tech. MTT-21, 99–104 (1973).
[CrossRef]

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

1972 (1)

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme, a l’étude des réseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[CrossRef]

1971 (3)

H. A. Kalhor and A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[CrossRef]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

1969 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

1966 (1)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B. 262, 468–471 (1966).

1956 (1)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[CrossRef]

1907 (1)

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

1897 (1)

Rayleigh (J. W. Strutt), “On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Phil. Mag. 44, 28–52 (1897).

Bates, R. H. T.

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–623 (1975).
[CrossRef]

Bolomey, J. C.

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section,” Proc. IEE 121, 794–804 (1974).

Cadilhac, M.

J. P. Hugonin, R. Petit, and M. Cadilhac, “On the use of plane wave expansions to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B. 262, 468–471 (1966).

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

Davies, J. B.

J. B. Davies, “A least-squares boundary residual method for the numerical solution of scattering problems,” IEEE Trans. Microwave Theory Tech. MTT-21, 99–104 (1973).
[CrossRef]

DeSanto, J. A.

J. A. DeSanto, “Theoretical methods in ocean acoustics,” in Ocean Acoustics (Springer-Verlag, Berlin, 1979), Chap. 2, p. 54.

Fokkema, J. T.

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface,” Radio Sci. 15, 723–732 (1980).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle,” IEEE Trans. Antennas Propag. AP-27, 577–583 (1979).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
[CrossRef]

Hugonin, J. P.

Ikuno, H.

H. Ikuno and K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

Kalhor, H. A.

H. A. Kalhor, “Numerical evaluation of Rayleigh hypothesis for analyzing scattering from corrugated gratings,” IEEE Trans. Antennas Propag. AP-24, 884–889 (1976).
[CrossRef]

H. A. Kalhor and A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[CrossRef]

Maystre, D.

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme, a l’étude des réseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[CrossRef]

Meecham, W. C.

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[CrossRef]

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

Neureuther, A. R.

H. A. Kalhor and A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[CrossRef]

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

Petit, R.

J. P. Hugonin, R. Petit, and M. Cadilhac, “On the use of plane wave expansions to describe the field diffracted by a grating,” J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme, a l’étude des réseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[CrossRef]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B. 262, 468–471 (1966).

Rayleigh,

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Rayleigh (J. W. Strutt), “On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Phil. Mag. 44, 28–52 (1897).

Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, London, 1896) (Dover, New York, 1945); The Theory of Sound, Vol.  II, pp. 89, 297–311.

van den Berg, P. M.

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface,” Radio Sci. 15, 723–732 (1980).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle,” IEEE Trans. Antennas Propag. AP-27, 577–583 (1979).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of reflection by a grating,” J. Opt. Soc. Am. 69, 27–31 (1979).
[CrossRef]

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

P. M. van den Berg, “Review of some computational techniques in scattering and diffraction,” in Proceedings International U.R.S.I.-Symposium 1980 (Union Radio-Scientifique Internationale, Brussels, 1980), pp. 211B/1–211B/6.

Wirgin, A.

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[CrossRef]

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section,” Proc. IEE 121, 794–804 (1974).

Yasuura, K.

H. Ikuno and K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzada, eds. (Union Radio-Scientifique Internationale, Brussels, 1971), pp. 257–270.

Zaki, K. A.

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

Appl. Sci. Res. (1)

P. M. van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).

C. R. Acad. Sci. B. (1)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. B. 262, 468–471 (1966).

IEEE Trans. Antennas Propag. (4)

K. A. Zaki and A. R. Neureuther, “Scattering from a perfectly conducting surface with a sinusoidal height profile,” IEEE Trans. Antennas Propag. AP-19, 208–214 (1971).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a cylindrical obstacle,” IEEE Trans. Antennas Propag. AP-27, 577–583 (1979).
[CrossRef]

H. Ikuno and K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

H. A. Kalhor, “Numerical evaluation of Rayleigh hypothesis for analyzing scattering from corrugated gratings,” IEEE Trans. Antennas Propag. AP-24, 884–889 (1976).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. B. Davies, “A least-squares boundary residual method for the numerical solution of scattering problems,” IEEE Trans. Microwave Theory Tech. MTT-21, 99–104 (1973).
[CrossRef]

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory Tech. MTT-23, 605–623 (1975).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. Wirgin, “Reflection from a corrugated surface,” J. Acoust. Soc. Am. 68, 692–699 (1980).
[CrossRef]

J. Appl. Phys. (1)

W. C. Meecham, “Variational method for the calculation of the distribution of energy reflected from a periodic surface,” J. Appl. Phys. 27, 361–367 (1956).
[CrossRef]

J. Opt. Soc. Am. (3)

Nouv. Rev. Opt. (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[CrossRef]

Phil. Mag. (1)

Rayleigh (J. W. Strutt), “On the incidence of aerial and electromagnetic waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Phil. Mag. 44, 28–52 (1897).

Proc. Cambridge Philos. Soc. (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface,” Proc. Cambridge Philos. Soc. 65, 773–791 (1969).
[CrossRef]

Proc. IEE (1)

J. C. Bolomey and A. Wirgin, “Numerical comparison of the Green’s function and the Waterman and Rayleigh theories of scattering from a cylinder with arbitrary cross-section,” Proc. IEE 121, 794–804 (1974).

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

Rayleigh (J. W. Strutt), “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

Radio Sci. (2)

R. F. Millar, “The Rayleigh hypothesis and a related least-square solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
[CrossRef]

P. M. van den Berg and J. T. Fokkema, “The Rayleigh hypothesis in the theory of diffraction by a perturbation in a plane surface,” Radio Sci. 15, 723–732 (1980).
[CrossRef]

Rev. Phys. Appl. (1)

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme, a l’étude des réseaux,” Rev. Phys. Appl. 7, 427–441 (1972).
[CrossRef]

Other (5)

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980).
[CrossRef]

K. Yasuura, “A view of numerical methods in diffraction problems,” in Progress in Radio Science 1966–1969, W. V. Tilston and M. Sauzada, eds. (Union Radio-Scientifique Internationale, Brussels, 1971), pp. 257–270.

P. M. van den Berg, “Review of some computational techniques in scattering and diffraction,” in Proceedings International U.R.S.I.-Symposium 1980 (Union Radio-Scientifique Internationale, Brussels, 1980), pp. 211B/1–211B/6.

Rayleigh (J. W. Strutt), The Theory of Sound, 2nd ed. (Macmillan, London, 1896) (Dover, New York, 1945); The Theory of Sound, Vol.  II, pp. 89, 297–311.

J. A. DeSanto, “Theoretical methods in ocean acoustics,” in Ocean Acoustics (Springer-Verlag, Berlin, 1979), Chap. 2, p. 54.

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Figures (5)

Fig. 1
Fig. 1

Grating configuration and incident wave. S1 denotes a single period of the domain zmax < z < ∞; S2 denotes a single period of the domain zmin < z < zmax (valley of the groove); ν denotes the normal vector to Λ.

Fig. 2
Fig. 2

Grating with a sinusoidal profile and normally incident wave.

Fig. 3
Fig. 3

The mean-square error in the boundary condition and the relative error in the power balance as a function of the number of equations (=2N + 1), using the point-matching technique in the Rayleigh method, for the cases of E and H polarization; h/D is a parameter.

Fig. 4
Fig. 4

The mean-square error in the boundary condition and the relative error in the power balance as a function of the number of equations (=2N + 1), using the Fourier technique in the Rayleigh method, for the cases of E and H polarization; h/D is a parameter.

Fig. 5
Fig. 5

The mean-square error in the boundary condition and the relative error in the power balance as a function of the number of equations (=2N + 1), using the least-square error technique in the Rayleigh method, for the cases of E and H polarization; h/D is a parameter.

Equations (16)

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

E i = { ( - γ 0 / ω 0 ) H 0 , E 0 , ( - α 0 / ω 0 ) H 0 } exp ( i α 0 x - i γ 0 z ) , H i = { ( + γ 0 / ω μ 0 ) E 0 , H 0 , ( + α 0 / ω μ 0 ) E 0 } exp ( i α 0 x - i γ 0 z ) ,
E 0 0 , H 0 = 0 for the case of E polarization , E 0 = 0 , H 0 0 for the case of H polarization ,
α 0 = k 0 sin θ 0 ,             γ 0 = k 0 cos θ 0 ,             k 0 = ω ( 0 μ 0 ) 1 / 2 .
{ E s , H s } = n A n + { e n + , h n + } ,             ( x , z ) S 1 ,
e n + = { ( + γ n / ω 0 ) H 0 , E 0 , ( - α n / ω 0 ) H 0 } exp ( i α n x + i γ n z ) , h n + = { ( - γ n / ω μ 0 ) E 0 , H 0 , ( + α n / ω μ 0 ) E 0 } exp ( i α n x + i γ n z ) ,
α n = α 0 + 2 π n / D , γ n = ( k 0 2 - α n 2 ) 1 / 2             with Re ( γ n ) 0 and Im ( γ n ) 0.
ν × E s = - ν × E i ,             r L ,
n A n R ( ν × e n + ) = - ν × E i ,             r L ,
n A n R ϕ m ,             e n + = - ϕ m , E i             for all m ,
f , g = L ( ν × f * ) · ( ν × g ) d s ,
A n R = A n + for all n .
E R R ( ν × E s ) = L | ν × E s - n = - N N A n + ( N ) ( ν × e n + ) | 2 d s 0             as N .
n = - N N A n + ( N ) e n + ( r ) E s ( r )             for any r ( S 1 S 2 ) as N .
n = - N N A n + ( N ) e m + , e n + = - e m + , E i ,             m = 0 , ± 1 , ± 2 , , ± N .
ERROR POWER BALANCE = | calculated reflected power - incident power incident power | = | propagating waves A n + ( N ) 2 γ n / γ 0 - 1 | ,
MEAN - SQUARE ERROR = L | n = - N N A n + ( N ) ( ν × e n + ) + ν × E i | 2 d s L ν × E i 2 d s .