Abstract

We recently introduced a method of variation of boundaries for the solution of diffraction problems [ J. Opt. Soc. Am. A 10, 1168 ( 1993)]. This method, which is based on a theorem of analyticity of the electromagnetic field with respect to variations of the interfaces, has been successfully applied in problems of diffraction of light by perfectly conducting gratings. We continue our investigation of diffraction problems. Using our previous results on analytic dependence with respect to the grating groove depth, we present a new numerical algorithm that applies to dielectric and metallic gratings. We also incorporate Padé approximation in our numerics. This addition enlarges the domain of applicability of our methods, and it results in computer codes that can predict more accurately the response of diffraction gratings in the resonance region. In many cases results are obtained that are several orders of magnitude more accurate than those given by other methods available at present, such as the integral or differential formalisms. We present a variety of numerical applications, including examples for several types of grating profile and for wavelengths of light ranging from microwaves to ultraviolet, and we compare our results with experimental data. We also use Padé approximants to gain insight into the analytic structure and the spectrum of singularities of the fields as functions of the groove depth. Finally, we discuss some connections between Padé approximation and another summation mechanism, enhanced convergence, which we introduced in the earlier paper. It is argued that, provided that certain numerical difficulties can be overcome, the performance of our algorithms could be further improved by a combination of these summation methods.

© 1993 Optical Society of America

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References

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  1. O. P. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
    [CrossRef]
  2. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
    [CrossRef]
  3. Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  4. J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
    [CrossRef]
  5. W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,” J. Rational Mechan. Anal. 5, 323–334 (1956).
  6. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
    [CrossRef]
  7. D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
    [CrossRef]
  8. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [CrossRef]
  9. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  10. E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
    [CrossRef]
  11. H. Ikuno, K. Yasuura, “Improved point-matching method with applications to scattering from a periodic surface,”IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [CrossRef]
  12. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, pp. 3–67.
    [CrossRef]
  13. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [CrossRef]
  14. G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Reading, Mass., 1981).
  15. G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Reading, Mass., 1981).
  16. C. Brezinski, “Procedures for estimating the error in Padé approximation,” Math. Computat. 53, 639–648 (1965).
  17. S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
    [CrossRef]
  18. P. Graves-Morris, “The numerical calculation of Padé approximants,” Lect. Notes Math. 765, 231–245 (1979).
    [CrossRef]
  19. P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).
  20. J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
    [CrossRef]
  21. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 63–100.
    [CrossRef]
  22. D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
    [CrossRef]
  23. D. Maystre, M. Nevibre, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 159–225.
    [CrossRef]
  24. R. Deleuil, “Réalisation et utilisation d’un appareillage destiné à l’étude des dioptres irreéguliers et des réseaux en ondes millimétriques,” Opt. Acta 16, 23–35 (1969).
    [CrossRef]
  25. M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
    [CrossRef]
  26. D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1981).
  27. O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Computat. (to be published).

1993 (1)

1992 (1)

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

1990 (2)

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
[CrossRef]

D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
[CrossRef]

1986 (1)

S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
[CrossRef]

1979 (1)

P. Graves-Morris, “The numerical calculation of Padé approximants,” Lect. Notes Math. 765, 231–245 (1979).
[CrossRef]

1978 (1)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
[CrossRef]

1973 (2)

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

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

1971 (3)

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

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

1970 (1)

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

1969 (1)

R. Deleuil, “Réalisation et utilisation d’un appareillage destiné à l’étude des dioptres irreéguliers et des réseaux en ondes millimétriques,” Opt. Acta 16, 23–35 (1969).
[CrossRef]

1965 (2)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

C. Brezinski, “Procedures for estimating the error in Padé approximation,” Math. Computat. 53, 639–648 (1965).

1956 (1)

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,” J. Rational Mechan. Anal. 5, 323–334 (1956).

Baker, G. A.

G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Reading, Mass., 1981).

G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Reading, Mass., 1981).

Bird, V. M.

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Bousquet, J.

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Brezinski, C.

C. Brezinski, “Procedures for estimating the error in Padé approximation,” Math. Computat. 53, 639–648 (1965).

Bruno, O. P.

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Computat. (to be published).

Cabay, S.

S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
[CrossRef]

Choi, D.

S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
[CrossRef]

Cox, J. A.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
[CrossRef]

Deleuil, R.

R. Deleuil, “Réalisation et utilisation d’un appareillage destiné à l’étude des dioptres irreéguliers et des réseaux en ondes millimétriques,” Opt. Acta 16, 23–35 (1969).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Graves-Morris, P.

P. Graves-Morris, “The numerical calculation of Padé approximants,” Lect. Notes Math. 765, 231–245 (1979).
[CrossRef]

G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Reading, Mass., 1981).

G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Reading, Mass., 1981).

Hutley, M. C.

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Ikuno, H.

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

Maystre, D.

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
[CrossRef]

D. Maystre, M. Nevibre, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 159–225.
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, pp. 3–67.
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

Meecham, W. C.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,” J. Rational Mechan. Anal. 5, 323–334 (1956).

Millar, R. F.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

Nevibre, M.

D. Maystre, M. Nevibre, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 159–225.
[CrossRef]

Nevière, M.

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
[CrossRef]

Pavageau, J.

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Petit, R.

D. Maystre, M. Nevibre, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 159–225.
[CrossRef]

Rayleigh,

Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

Reitich, F.

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries,” J. Opt. Soc. Am. A 10, 1168–1175 (1993).
[CrossRef]

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Computat. (to be published).

Talbot, D.

D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
[CrossRef]

Titchener, J. B.

D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
[CrossRef]

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Van den Berg, P. M.

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

Wait, J. R.

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

Willis, J. R.

D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
[CrossRef]

Yasuura, K.

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

Ann. Phys. (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Appl. Sci. Res. (1)

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

IEEE Trans. Antennas Propag. (1)

H. Ikuno, 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. Opt. (1)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. 9, 301–306 (1978).
[CrossRef]

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

J. Rational Mechan. Anal. (1)

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,” J. Rational Mechan. Anal. 5, 323–334 (1956).

Lect. Notes Math. (1)

P. Graves-Morris, “The numerical calculation of Padé approximants,” Lect. Notes Math. 765, 231–245 (1979).
[CrossRef]

Math. Computat. (1)

C. Brezinski, “Procedures for estimating the error in Padé approximation,” Math. Computat. 53, 639–648 (1965).

Opt. Acta (3)

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

R. Deleuil, “Réalisation et utilisation d’un appareillage destiné à l’étude des dioptres irreéguliers et des réseaux en ondes millimétriques,” Opt. Acta 16, 23–35 (1969).
[CrossRef]

M. C. Hutley, V. M. Bird, “A detailed experimental study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 20, 771–782 (1973).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[CrossRef]

Proc. R. Soc. Edinburgh Sect. A (1)

O. P. Bruno, F. Reitich, “Solution of a boundary value problem for Helmholtz equation via variation of the boundary into the complex domain,” Proc. R. Soc. Edinburgh Sect. A 122, 317–340 (1992).
[CrossRef]

Radio Sci. (1)

J. R. Wait, “Perturbation analysis for reflection from two-dimensional periodic sea waves,” Radio Sci. 6, 387–391 (1971).
[CrossRef]

SIAM J. Comput. (1)

S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
[CrossRef]

Wave Motion (1)

D. Talbot, J. B. Titchener, J. R. Willis, “The reflection of electromagnetic waves from very rough interfaces,” Wave Motion 12, 245–260 (1990).
[CrossRef]

Other (10)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Reading, Mass., 1981).

G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Reading, Mass., 1981).

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1984), Vol. 21, pp. 3–67.
[CrossRef]

D. E. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1981).

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” Math. Computat. (to be published).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

D. C. Dobson, J. A. Cox, “An integral equation method for biperiodic diffraction structures,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1545, 106–113 (1991).
[CrossRef]

D. Maystre, M. Nevibre, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 159–225.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Efficiency curves for a perfectly conducting ruled grating (blaze angle 37°, included angle 94°, deviation angle 8.9°) in the microwave region: (a) TE polarization, (b) TM polarization.

Fig. 2
Fig. 2

Efficiency curves of a 26.75° blaze angle perfectly conducting echelette grating in the infrared (angular deviation 3.5° between incident and −1-order diffracted waves): (a) TE polarization, (b) TM polarization.

Fig. 3
Fig. 3

Efficiency of an 830-groove/mm sinusoidal holographic silver grating in the visible range (λ = 0.521 μm) as a function of the incidence. Dashed curve, TE polarization; solid curve, TM polarization.

Fig. 4
Fig. 4

TE efficiency curve for a 158-groove/mm aluminum ruled grating (blaze angle 1.66°, included angle 90°, deviation angle 3.5°) in the near UV.

Fig. 5
Fig. 5

Poles (○) and zeros (×) of the Padé approximants of B1(δ): (a) [28/28] approximant, (b) [48/48] approximant.

Fig. 6
Fig. 6

C is the region of analyticity of the Rayleigh coefficients Br±(δ), and L is the lens-shaped region that is conformally transformed onto the right-half plane by means of g(δ) = [(Aδ)/(A + δ)]α. See text for a definition of the other parameters.

Fig. 7
Fig. 7

Plot of |g(δ = 0.1) − 1|/|g(δj) − 1| as functions of σ.

Tables (6)

Tables Icon

Table 1 Efficiencies for a Perfectly Conducting Sinusoidal Grating under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.4368: [32/32] Padé Approximants, TE Polarization

Tables Icon

Table 2 Efficiencies for a Perfectly Conducting Sinusoidal Grating under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.4368: [32/32] Padé Approximants, TM Polarization

Tables Icon

Table 3 Efficiencies for a Perfectly Conducting Symmetric Echelette Grating under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.4368: [12/12] Padé Approximants, TE Polarizationa

Tables Icon

Table 4 Efficiencies for a Perfectly Conducting Symmetric Echelette Grating under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.4368: [12/12] Padé Approximants, TM Polarizationa

Tables Icon

Table 5 Reflected and Transmitted Energies for a Sinusoidal Grating with Index of Refraction ν0 = 2, under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83: [20/20] Padé Approximants, TE Polarizationa

Tables Icon

Table 6 Reflected and Transmitted Energies for a Sinusoidal Grating with Index of Refraction ν0 = 2, under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83: [20/20] Padé Approximants, TM Polarizationa

Equations (57)

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y = f δ ( x ) = δ f ( x ) ,
E i = A exp ( i α x - i β y ) exp ( - i ω t ) , H i = B exp ( i α x - i β y ) exp ( i ω t ) .
Δ u ± + ( k ± ) 2 u ± = 0             in Ω ± = { ± y > ± δ f ( x ) }
u + - u - = - exp [ i a x - i β δ f ( x ) ]             on y = δ f ( x ) , u + n δ - C 0 2 u - n δ = - n δ { exp [ i α x - i β δ f ( x ) ] }             on y = δ f ( x )
C 0 = { 1 for TE polarization ( k + / k - ) for TM polarization ,
K = 2 π d ,             α n = α + n K ,             α n 2 + ( β n ± ) 2 = ( k ± ) 2 ,
u + = n = - B n + exp ( i α n x + i β n + y ) .
u - = n = - B n - exp ( i α n x - i β n - y ) ,
n U + β n + B n + 2 + C 0 2 n U - β n - B n - 2 = β 0 + ( = β ) ,
U ± { n : β n ± > 0 } .
n U + e n + + C 0 2 n U - e n - = 1
e n ± = β n ± B n ± 2 / β 0 +
n U + e n = 1 .
u ± ( x , δ f ( x ) , δ ) , u ± n δ ( x , δ f ( x ) ) ( x , δ f ( x ) , δ )
u ± ( x , y , δ ) = n = 0 u n ± ( x , y ) δ n ,
Δ u n ± + ( k ± ) 2 u n ± = 0             in { ( x , y ) : ± y > 0 }
k = 0 n f ( x ) n - k ( n - k ) ! [ n - k y n - k ( 1 k ! k u + δ k ) ( x , 0 , 0 ) - n - k y n - k ( 1 k ! k u - δ k ) ( x , 0 , 0 ) ] = - ( - i β ) n n ! f ( x ) n exp ( i α x ) .
n δ = 1 [ 1 + δ 2 f ( x ) 2 ] 1 / 2 ( - δ f ( x ) , 1 ) ,
- δ f ( x ) ( u + x - C 0 2 u - x ) + ( u + y - C 0 2 u - y ) = [ δ f ( x ) i α + i β ] exp [ i α x - i β δ f ( x ) ] .
k = 0 n f ( x ) n - k ( n - k ) ! [ n - k + 1 y n - k + 1 ( 1 k ! k u + δ k ) ( x , 0 , 0 ) - C 0 2 n - k + 1 y n - k + 1 ( 1 k ! k u - δ k ) ( x , 0 , 0 ) ] - k = 0 n - 1 f ( x ) f ( x ) n - k - 1 ( n - k - 1 ) ! [ n - k y n - k - 1 x ( 1 k ! k u + δ k ) ( x , 0 , 0 ) - C 0 2 n - k y n - k - 1 x ( 1 k ! k u - δ k ) ( x , 0 , 0 ) ] = 1 n ! [ i α n ( - i β ) n - 1 f ( x ) f ( x ) n - 1 - ( - i β ) n + 1 f ( x ) n ] exp ( i α x ) .
u k ± ( x , y ) = 1 k ! k u ± δ k ( x , y , 0 ) ,
u n + = u n - = - ( - i β ) n n ! f n exp ( i α x ) - k = 0 n - 1 f n - k ( n - k ) ! ( n - k u k + y n - k - n - k u k - y n - k ) ,
u n + y - C 0 2 u n - y = 1 n ! [ i α n ( - i β ) n - 1 ( f f n - 1 ) - ( - i β ) n + 1 f n ] × exp ( i α x ) + k = 0 n - 1 ( f f n - k - 1 ) ( n - k - 1 ) ! × [ n - k u k + x y n - k - 1 - C 0 2 n - k u k - x y n - k - 1 ] - k = 0 n - 1 f n - k ( n - k ) ! × ( n - k + 1 u k + y n - k + 1 - C 0 2 n - k + 1 u k - y n - k + 1 ) .
u n ± ( x , y ) = r = - d n , r ± exp ( i α r x ± i β r ± y )
u ± ( x , y , δ ) = r = - B r ± ( δ ) exp ( i α r x ± i β r ± y ) .
d n , r ± = 1 n ! d n B r ± d δ n ( 0 ) ,
B r ± ( δ ) = n = 0 d n , r ± δ n
f ( x ) = r = - F F C 1 , r exp ( i K r x ) ,             K = 2 π d
f ( x ) l l ! = r = - l F l F C l , r exp ( i K r x ) .
r = - ( d n , r + - d n , r - ) exp ( i α r x ) = - ( - i β ) n r = - n F n F C n , r exp ( i α r x ) - k = 0 n - 1 [ r = - ( n - k ) F ( n - k ) F C n - k , r exp ( i K r x ) ] × { r = - [ ( i β r + ) n - k d k , r + - ( - i β r - ) n - k d k , r - ] exp ( i α r x ) } ,
1 n ! [ n f ( x ) f ( x ) n - 1 ] = d d x ( f n n ! ) = r = - n F n F C n , r i K r exp ( i K r x ) ,
r = - ( i β r + d n , r + + C 0 2 i β r - d n , r - ) exp ( i α r x ) = r = - n F n F C n , r [ i α ( - i β ) n - 1 ( i K r ) - ( - i β ) n + 1 ] exp ( i α r x ) + k = 0 n - 1 [ r = - ( n - k ) F ( n - k ) F C n - k , r i K r exp ( i K r x ) ] × { r = - [ ( i β r + ) n - k - 1 ( i α r ) d k , r + - C 0 2 ( - i β r - ) n - k - 1 ( i α r ) d k , r - ] × exp ( i α r x ) } - k = 0 n - 1 [ r = - ( n - k ) F ( n - k ) F C n - k , r exp ( i K r x ) ] × { r = - [ ( i β r + ) n - k + 1 d k , r + - C 0 2 ( - i β r - ) n - k + 1 d k , r - ] × exp ( i α r x ) } .
- r = - n F n F ( - i β ) n C n , r exp ( i α r x ) - k = 0 n - 1 q = - p = - ( n - k ) F ( n - k ) F C n - k , p [ ( i β q + ) n - k d k , q + - ( - i β q - ) n - k d k , q - ] × exp ( i K p x ) exp ( i α q x ) .
exp ( i K p x ) exp ( i α q x ) = exp ( i α p + q x ) ,
- r = - n F n F ( - i β ) n C n , r exp ( i α r x ) - k = 0 n - 1 q = - r = q - ( n - k ) F q + ( n - k ) F C n - k , r - q × [ ( i β q + ) n - k d k , q + - ( - i β q - ) n - k d k , q - ] exp ( i α r x ) = - r = - n F n F ( - i β ) n C n , r exp ( i α r x ) - r = - { k = 0 n - 1 q = r - ( n - k ) F r + ( n - k ) F C n - k , r - q × [ ( i β q + ) n - k d k , q + - ( - i β q - ) n - k d k , q - ] } exp ( i α r x ) .
d k , q ± = 0 ,             if q > k F .
r = - n F n F ( d n , r + - d n , r - ) exp ( i α r x ) = - r = - n F n F { ( - i β ) n C n , r + k = 0 n - 1 q = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] C n - k , r - q × [ ( i β q + ) n - k d k , q + - ( - i β q - ) n - k d k , q - ] } exp ( i α r x ) .
r = - n F n F ( i β r + d n , r + + C 0 2 i β r - d n , r - ) exp ( i α r x ) = r = - n F n F ( C n , r ( - i β ) n - 1 [ ( i α ) ( i K r ) - ( - i β ) 2 ] + k = 0 n - 1 q = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] C n - k , r - q { [ i K ( r - q ) ] ( i α q ) × [ ( i β q + ) n - k - 1 d k , q + - C 0 2 ( - i β q - ) n - k - 1 d k , q - ] - [ ( i β q + ) n - k + 1 d k , q + - C 0 2 ( - i β q - ) n - k + 1 d k , q - ] } ) exp ( i α r x ) .
d n , r + - d n , r - = - ( - i β ) n C n , r - k = 0 n - 1 q = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] C n - k , r - q × [ ( i β q + ) n - k d k , q + - ( - i β q - ) n - k d k , q - ] ,
i β r + d n , r + + C 0 2 i β r - d n , r - = C n , r ( - i β ) n - 1 [ ( i α ) ( i K r ) - ( - i β ) 2 ] + k = 0 n - 1 q = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] C n - k , r - q { [ i K ( r - q ) ] × ( i α q ) [ ( i β q + ) n - k - 1 d k , q + - C 0 2 ( - i β q - ) n - k - d k , q - ] - [ ( i β q + ) n - k + 1 d k , q + - C 0 2 ( - i β q - ) n - k + 1 d k , q - ] } .
max [ - k F , r - ( n - k ) F ] q min [ k F , r + ( n - k ) F ] .
B ( δ ) = n = 0 d n δ n
[ L / M ] = a 0 + a 1 δ + + a L δ L 1 + b 1 δ + + b M δ M ,
n = 0 d n δ n = a 0 + a 1 δ + + a L δ L 1 + b 1 δ + + b M δ M + O ( δ L + M + 1 ) ,
[ d L - M + 1 d L - M + 2 d L d L - M + 2 d L - M + 3 d L + 1 d L d L + 1 d L + M - 1 ] [ b M b M - 1 b 1 ] = - [ d L + 1 d L + 2 d L + M ] .
f ( x ) = h 2 cos ( 2 π x / d ) = h 2 cos ( K x )
= 1 - n U + e n .
y = h 2 g ( 2 π x / d ) ,
g ( x ) = { - 2 x π - 2 if - π x - π 2 2 x π if - π 2 x π 2 - 2 x π + 2 if π 2 x π .
r = - F F C 1 , r exp ( i K r x ) ,
R = 0.117274 , T = 0.882759 , = 1 - ( T + R ) = 3.3 × 10 - 5 ;
f δ ( x ) = δ [ exp ( i 2 π x ) + exp ( - i 2 π x ) ] = 2 δ cos ( 2 π x )
h / d = 4 δ .
ξ = g ( δ ) ,
ξ = g ( δ ) = ( A - δ A + δ ) α ,
B r [ g - 1 ( ξ ) ] .
max j [ g ( δ ) - 1 g ( δ j ) - 1 ]

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