K. Yasuura and Y. Okuno, "Singular-smoothing procedure on Fourier analysis," Mem. Fac. Eng. Kyushu Univ. 41, 123–141 (1981).

J. P. Hugonin, R. Petit, and M. Cadilhac, "Plane wave expansions used to describe the field diffracted by a grating," J. Opt. Soc. Am. 71, 593–598 (1981).

P. M. van den Berg, "Reflection by a grating: Rayleigh methods," J. Opt. Soc. Am. 71, 1224–1229 (1981).

H. Ikuno and K. Yasuura, "Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method," Radio Sci. 13, 937–946 (1978).

The higher-order SP means the technique with higher-order integration by parts. The mathematical foundations of the higher-order SP can be found in K. Yasuura, Y. Okuno, and H. Ikuno, "Smoothing process on series expansion of functions," Kyushu Univ. Tech. Rep. 50, 463–469 (1978) (in Japanese). The method presented in the above article can be applied for the problem with sufficiently smooth boundary. For the present case, in which the boundary has edge points, the higher-order SP must be by means of the SSP whose theory is given in Ref. 6, because the higher-order derivatives of the Green function ∂_{ss}^{2}*G*(*P*, *s*), ∂_{sss}^{3}*G*(*P*, *s*),... are no longer square integrable on *L*.

K. Yasuura, Y. Okuno, and H. Ikuno, "Numerical analysis of echelette grating—smoothing process on mode-matching method," Trans. Inst. Electron. Commun. Eng. Jpn. 60-B, 189–196 (1977) (in Japanese). The smoothing procedure in a primitive form was presented in the paper cited. The authors have made some refinements of the theory of the procedure and, consequently, the numerical algorithm explained in the present paper was developed.

H. Ikuno and K. Yasuura, "Improved point matching method with application to scattering from a periodic surface," IEEE Trans. Antennas Propag. AP-21, 657–662 (1973). The numerical technique employed in this paper was named the IPMM because the conventional point-matching method was just improved by doubling the number of testing points. The convergence of the IPMM solution can be proved even if Rayleigh's infinite series fails to converge.^{3}

The higher-order SP means the technique with higher-order integration by parts. The mathematical foundations of the higher-order SP can be found in K. Yasuura, Y. Okuno, and H. Ikuno, "Smoothing process on series expansion of functions," Kyushu Univ. Tech. Rep. 50, 463–469 (1978) (in Japanese). The method presented in the above article can be applied for the problem with sufficiently smooth boundary. For the present case, in which the boundary has edge points, the higher-order SP must be by means of the SSP whose theory is given in Ref. 6, because the higher-order derivatives of the Green function ∂_{ss}^{2}*G*(*P*, *s*), ∂_{sss}^{3}*G*(*P*, *s*),... are no longer square integrable on *L*.

H. Ikuno and K. Yasuura, "Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method," Radio Sci. 13, 937–946 (1978).

K. Yasuura, Y. Okuno, and H. Ikuno, "Numerical analysis of echelette grating—smoothing process on mode-matching method," Trans. Inst. Electron. Commun. Eng. Jpn. 60-B, 189–196 (1977) (in Japanese). The smoothing procedure in a primitive form was presented in the paper cited. The authors have made some refinements of the theory of the procedure and, consequently, the numerical algorithm explained in the present paper was developed.

H. Ikuno and K. Yasuura, "Improved point matching method with application to scattering from a periodic surface," IEEE Trans. Antennas Propag. AP-21, 657–662 (1973). The numerical technique employed in this paper was named the IPMM because the conventional point-matching method was just improved by doubling the number of testing points. The convergence of the IPMM solution can be proved even if Rayleigh's infinite series fails to converge.^{3}

K. Yasuura and T. Itakura, "Approximation method for wave function," Kyushu Univ. Tech. Rep. 38, 72–77 (1965); 38, 378–385 (1966); 39, 51–56 (1966) (in Japanese). This paper deals with the fundamental theory of the CMMM including the completeness theorem of the set of the modal functions; its contents are briefly presented in K. Yasuura, "A view of numerical methods in diffraction problems," in Progress in Radio Science, W. V. Tilson and M. Sauzade, eds. (International Union of Radio Science, Brussels, 1971), pp. 257–270.

K. Yasuura and Y. Okuno, "Singular-smoothing procedure on Fourier analysis," Mem. Fac. Eng. Kyushu Univ. 41, 123–141 (1981).

The higher-order SP means the technique with higher-order integration by parts. The mathematical foundations of the higher-order SP can be found in K. Yasuura, Y. Okuno, and H. Ikuno, "Smoothing process on series expansion of functions," Kyushu Univ. Tech. Rep. 50, 463–469 (1978) (in Japanese). The method presented in the above article can be applied for the problem with sufficiently smooth boundary. For the present case, in which the boundary has edge points, the higher-order SP must be by means of the SSP whose theory is given in Ref. 6, because the higher-order derivatives of the Green function ∂_{ss}^{2}*G*(*P*, *s*), ∂_{sss}^{3}*G*(*P*, *s*),... are no longer square integrable on *L*.

K. Yasuura, Y. Okuno, and H. Ikuno, "Numerical analysis of echelette grating—smoothing process on mode-matching method," Trans. Inst. Electron. Commun. Eng. Jpn. 60-B, 189–196 (1977) (in Japanese). The smoothing procedure in a primitive form was presented in the paper cited. The authors have made some refinements of the theory of the procedure and, consequently, the numerical algorithm explained in the present paper was developed.

Y. Okuno and K. Yasuura, "Numerical algorithm based on the mode-matching method with singular-smoothing procedure for analysing edge-type scattering problems," IEEE Trans. Antennas Propag. (to be published).

K. Yasuura and Y. Okuno, "Singular-smoothing procedure on Fourier analysis," Mem. Fac. Eng. Kyushu Univ. 41, 123–141 (1981).

H. Ikuno and K. Yasuura, "Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method," Radio Sci. 13, 937–946 (1978).

The higher-order SP means the technique with higher-order integration by parts. The mathematical foundations of the higher-order SP can be found in K. Yasuura, Y. Okuno, and H. Ikuno, "Smoothing process on series expansion of functions," Kyushu Univ. Tech. Rep. 50, 463–469 (1978) (in Japanese). The method presented in the above article can be applied for the problem with sufficiently smooth boundary. For the present case, in which the boundary has edge points, the higher-order SP must be by means of the SSP whose theory is given in Ref. 6, because the higher-order derivatives of the Green function ∂_{ss}^{2}*G*(*P*, *s*), ∂_{sss}^{3}*G*(*P*, *s*),... are no longer square integrable on *L*.

K. Yasuura, Y. Okuno, and H. Ikuno, "Numerical analysis of echelette grating—smoothing process on mode-matching method," Trans. Inst. Electron. Commun. Eng. Jpn. 60-B, 189–196 (1977) (in Japanese). The smoothing procedure in a primitive form was presented in the paper cited. The authors have made some refinements of the theory of the procedure and, consequently, the numerical algorithm explained in the present paper was developed.

H. Ikuno and K. Yasuura, "Improved point matching method with application to scattering from a periodic surface," IEEE Trans. Antennas Propag. AP-21, 657–662 (1973). The numerical technique employed in this paper was named the IPMM because the conventional point-matching method was just improved by doubling the number of testing points. The convergence of the IPMM solution can be proved even if Rayleigh's infinite series fails to converge.^{3}

Y. Okuno and K. Yasuura, "Numerical algorithm based on the mode-matching method with singular-smoothing procedure for analysing edge-type scattering problems," IEEE Trans. Antennas Propag. (to be published).

K. Yasuura and T. Itakura, "Approximation method for wave function," Kyushu Univ. Tech. Rep. 38, 72–77 (1965); 38, 378–385 (1966); 39, 51–56 (1966) (in Japanese). This paper deals with the fundamental theory of the CMMM including the completeness theorem of the set of the modal functions; its contents are briefly presented in K. Yasuura, "A view of numerical methods in diffraction problems," in Progress in Radio Science, W. V. Tilson and M. Sauzade, eds. (International Union of Radio Science, Brussels, 1971), pp. 257–270.

H. Ikuno and K. Yasuura, "Improved point matching method with application to scattering from a periodic surface," IEEE Trans. Antennas Propag. AP-21, 657–662 (1973). The numerical technique employed in this paper was named the IPMM because the conventional point-matching method was just improved by doubling the number of testing points. The convergence of the IPMM solution can be proved even if Rayleigh's infinite series fails to converge.^{3}

The higher-order SP means the technique with higher-order integration by parts. The mathematical foundations of the higher-order SP can be found in K. Yasuura, Y. Okuno, and H. Ikuno, "Smoothing process on series expansion of functions," Kyushu Univ. Tech. Rep. 50, 463–469 (1978) (in Japanese). The method presented in the above article can be applied for the problem with sufficiently smooth boundary. For the present case, in which the boundary has edge points, the higher-order SP must be by means of the SSP whose theory is given in Ref. 6, because the higher-order derivatives of the Green function ∂_{ss}^{2}*G*(*P*, *s*), ∂_{sss}^{3}*G*(*P*, *s*),... are no longer square integrable on *L*.

K. Yasuura and Y. Okuno, "Singular-smoothing procedure on Fourier analysis," Mem. Fac. Eng. Kyushu Univ. 41, 123–141 (1981).

H. Ikuno and K. Yasuura, "Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method," Radio Sci. 13, 937–946 (1978).

K. Yasuura, Y. Okuno, and H. Ikuno, "Numerical analysis of echelette grating—smoothing process on mode-matching method," Trans. Inst. Electron. Commun. Eng. Jpn. 60-B, 189–196 (1977) (in Japanese). The smoothing procedure in a primitive form was presented in the paper cited. The authors have made some refinements of the theory of the procedure and, consequently, the numerical algorithm explained in the present paper was developed.

Interested readers can find the references in R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).

K. Yasuura and T. Itakura, "Approximation method for wave function," Kyushu Univ. Tech. Rep. 38, 72–77 (1965); 38, 378–385 (1966); 39, 51–56 (1966) (in Japanese). This paper deals with the fundamental theory of the CMMM including the completeness theorem of the set of the modal functions; its contents are briefly presented in K. Yasuura, "A view of numerical methods in diffraction problems," in Progress in Radio Science, W. V. Tilson and M. Sauzade, eds. (International Union of Radio Science, Brussels, 1971), pp. 257–270.

Y. Okuno and K. Yasuura, "Numerical algorithm based on the mode-matching method with singular-smoothing procedure for analysing edge-type scattering problems," IEEE Trans. Antennas Propag. (to be published).

The theory of the CMMM was presented in 1965–1966,^{3} although it appeared in an American journal in 1973 for the first time.^{2}

The completeness of {∂_{n}Φ_{µ}(*s*); µ = 0, ±1, ±2,...} holds excluding the discrete values of *kD* for which *k*_{yµ} = 0 (cutoff of the µth-order spectral mode) occurs. The proof of the completeness was presented in the general form in Ref. 3.

The speed of the convergence of the SP solutions was examined theoretically in Ref. 5, and the fact was made clear that the sequence of the SP solutions converges to the true solution more rapidly than the CMMM solutions.

The proof of this lemma was given in Ref. 5. The positive integer *N*_{0} that appeared in Eq. (39) is defined as follows: Since constants are square integrable on *L*, there is at least one element ∂_{n}Φ_{v0}(*s*) that is not orthogonal to constants in the complete set {∂;_{n}Φ_{µ} (*s*)}: (1, ∂_{n}Φ_{v0}) ≠ 0. There may be numerous ν_{0}'s for which this condition holds. Then, putting *N*_{0} = min{|ν_{0}|}, we have the integer *N*_{0} of Eq. (39).

If the cross section of the grating is represented in terms of a finite power series of *x*', we can evaluate the inner products analytically by the technique of integration by parts. Otherwise they must be calculated numerically by the IPMM technique; the number of divisions is equal to 2(2*N* = 1).