Abstract

We present a new numerical method for the solution of the problem of diffraction of light by a doubly periodic surface. This method is based on a high-order rigorous perturbative technique, whose application to singly periodic gratings was treated in the first two papers of this series [ J. Opt. Soc. Am. A 10, 1168, 2307 ( 1993)]. We briefly discuss the theoretical basis of our algorithm, namely, the property of analyticity of the diffracted fields with respect to variations of the interfaces. While the algebraic derivation of some basic recursive formulas is somewhat involved, it results in expressions that are easy to implement numerically. We present a variety of numerical examples (for bisinusoidal gratings) in order to demonstrate the accuracy exhibited by our method as well as its limited requirements in terms of computing power. Generalization of our computer code to crossed gratings other than bisinusoidal is in principle immediate, but the full domain of applicability of our algorithm remains to be explored. Comparison with results presented previously for actual experimental configurations shows a substantial improvement in the resolution of our numerics over that given by other methods introduced in the past.

© 1993 Optical Society of America

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References

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  1. 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]
  2. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Finitely conducting gratings, Padé approximants, and singularities,” J. Opt. Soc. Am. A 10, 2307–2316 (1993).
    [Crossref]
  3. C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
    [Crossref]
  4. P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [Crossref]
  5. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,”J. Opt. (Paris) 9, 301–306 (1978).
    [Crossref]
  6. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [Crossref]
  7. J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
    [Crossref]
  8. 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]
  9. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 227–276.
    [Crossref]
  10. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
    [Crossref]
  11. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
  12. 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]
  13. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [Crossref]
  14. 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 122A, 317–340 (1992).
    [Crossref]
  15. D. C. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,”J. Math. Anal. Appl. 166, 507–528 (1992).
    [Crossref]
  16. 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]
  17. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
    [Crossref]
  18. G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Boston, Mass., 1981).
  19. G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Boston, Mass., 1981).
  20. C. Brezinski, “Procedures for estimating the error in Padé approximation,” Math. Comp. 53, 639–648 (1965).
  21. S. Cabay, D. Choi, “Algebraic computations of scaled Padé fractions,” SIAM J. Comput. 15, 243–270 (1986).
    [Crossref]
  22. P. Graves-Morris, “The numerical calculation of Padé approximants,” Lect. Notes Math. 765, 231–245 (1979).
    [Crossref]
  23. E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic, Orlando, Fla., 1985).
  24. 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]

1993 (2)

1992 (2)

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 122A, 317–340 (1992).
[Crossref]

D. C. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,”J. Math. Anal. Appl. 166, 507–528 (1992).
[Crossref]

1990 (1)

1986 (1)

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

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

1979 (2)

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

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

1978 (2)

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[Crossref]

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

1976 (1)

J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
[Crossref]

1974 (1)

C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
[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. Comp. 53, 639–648 (1965).

Bachner, F. J.

J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
[Crossref]

Baker, G. A.

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

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

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 227–276.
[Crossref]

Brezinski, C.

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

Bruno, O. P.

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]

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 227–276.
[Crossref]

Dobson, D. C.

D. C. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,”J. Math. Anal. Appl. 166, 507–528 (1992).
[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]

Fan, J. C. C.

J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
[Crossref]

Friedman, A.

D. C. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,”J. Math. Anal. Appl. 166, 507–528 (1992).
[Crossref]

Gaylord, T. K.

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]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

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, Boston, Mass., 1981).

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

Horwitz, C. M.

C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
[Crossref]

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

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

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 227–276.
[Crossref]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
[Crossref]

Murphy, R. A.

J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
[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.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

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

Petit, R.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
[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]

Reitich, F.

Uretsky, J. L.

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

Vincent, P.

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[Crossref]

Ann. Phys. (1)

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

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[Crossref]

Appl. Phys. Lett. (1)

J. C. C. Fan, F. J. Bachner, R. A. Murphy, “Thin-film conducting microgrids as transparent heat mirrors,” Appl. Phys. Lett. 28, 440–442 (1976).
[Crossref]

J. Math. Anal. Appl. (1)

D. C. Dobson, A. Friedman, “The time-harmonic Maxwell equations in a doubly periodic structure,”J. Math. Anal. Appl. 166, 507–528 (1992).
[Crossref]

J. Opt. (Paris) (1)

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

J. Opt. Soc. Am. A (3)

Lect. Notes Math. (1)

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

Math. Comp. (1)

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

Opt. Commun. (2)

C. M. Horwitz, “A new solar selective surface,” Opt. Commun. 11, 210–212 (1974).
[Crossref]

P. Vincent, “A finite difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[Crossref]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Proc. R. Soc. Edinburgh (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 122A, 317–340 (1992).
[Crossref]

SIAM J. Comput. (1)

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

Other (9)

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]

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
[Crossref]

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

G. A. Baker, P. Graves-Morris, Padé Approximants. Part II: Extensions and Applications (Addison-Wesley, Boston, 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]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 227–276.
[Crossref]

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 1–52.
[Crossref]

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

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 (3)

Fig. 1
Fig. 1

E. R., Energy reflected by a sinusoidal grating in gold used with normally incident light of 0.65-μm wavelength. Curve 1, h = 0.040 μm; curve 2, h = 0.055 μm; curve 3, h = 0.070 μm: [6/6] Padé approximants.

Fig. 2
Fig. 2

Zeroth-order efficiency for a sinusoidal grating in gold having a groove depth h = 0.080 μm and a period of 0.60 μm, used with normally incident light: [6/6] Padé approximants.

Fig. 3
Fig. 3

Energy absorbed by a sinusoidal grating in copper having a groove depth h = 0.20 μm as a function of the wavelength for normally incident light: a, d = 0.7071 μm; b, d = 0.50 μm; c, d = 0.35 μm; d, d = 0.20 μm: [6/6] Padé approximants.

Tables (5)

Tables Icon

Table 1 Efficiencies for the Perfectly Conducting Sinusoidal Grating [Eq. (31)] under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83: [14/14] Padé Approximants

Tables Icon

Table 2 Efficiency of Order (0, 0) for the Perfectly Conducting Grating [Eq. (31)] under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.4368: [14/14] Padé Approximants

Tables Icon

Table 3 Efficiencies for the Sinusoidal Grating [Eq. (31)] with Index of Refraction ν0 = 2 under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83: [14/14] Padé Approximants

Tables Icon

Table 4 Convergence Study of the Reflected Energy for the Example in Fig. 1 (Gold) with Period Fixed at 0.62 μm and Wavelength at 0.65 μm: [ n - 1 2 / n - 1 2 ] Padé Approximants

Tables Icon

Table 5 Convergence Study of the Absorbed Energy for the Example in Figs. 3a and 3b (Copper) with Wavelength Fixed at λ = 0.3 μm and Period at d = 0.7071 μm for Fig. 3a and at d = 0.5000 μm for Fig. 3b: [ n - 1 2 / n - 1 2 ] Padé Approximants

Equations (64)

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

z = f ( x , y )
E i = A exp [ i ( α x + β y - γ z - i ω t ) ] , H i = B exp [ i ( α x + β y - γ z - i ω t ) ]
A · k = 0 ,             B = 1 ω μ 0 k × A ,
k = ( α , β , - γ )
× E = i ω μ 0 H , · E = 0 , × H = - i ω E , · H = 0.
Δ v + k 2 v = 0 ,
E up = E i + E + ,             H up = H i + H + , E down = E - ,             H down = H -
n × ( E up - E down ) = 0 ,             n × ( H up - H down ) = 0             on z = f ( x , y ) .
n × E up = 0             on z = f ( x , y ) .
v ( x + d 1 , y , z ) = exp ( i α d 1 ) v ( x , y , z ) , v ( x , y + d 2 , z ) = exp ( i β d 2 ) v ( x , y , z ) ,
E ± = r = - s = - B r , s ± exp ( i α r x + i β s y ± i γ r , s ± z ) ,
α r = α + r K 1 , β s = β + s K 2 , α r 2 + β s 2 + ( γ r , s ± ) 2 = ( k ± ) 2 ,
( k ± ) 2 = ω 2 ± μ 0 , K 1 = 2 π d 1 ,             K 2 = 2 π d 2 .
e r , s ± = B r , s ± 2 γ r , s ± γ 0 , 0 +
( r , s ) U + e r , s + + ( r , s ) U - e r , s - = 1 ,
z = δ f ( x , y )
E ( x , y , z ) = F [ i ω μ j Φ - j × Φ + 1 i ω Div ( j ) Φ ] d σ ( x , y , z ) , H ( x , y , z ) = F [ i ω j Φ + j × Φ + 1 i ω μ Div ( j ) Φ ] d σ ( x , y , z ) ,
E ± ( x , y , z ; δ ) = - μ 0 D δ ± ( j ) + ± i ω μ 0 S δ ± ( j ) - 1 i ω T δ ± ( j ) , H ± ( x , y , z ; δ ) = ± D δ ± ( j ) + ± i ω μ 0 S δ ± ( j ) - 1 i ω T δ ± ( j ) ,
D δ ± ( j ) ( x , y , z ) = × 0 d 1 0 d 2 Φ ± [ x - x , y - y , z - δ f ( x , y ) ] j ( x , y ) × { 1 + [ δ f x ( x , y ) ] 2 + [ δ f y ( x , y ) ] 2 } 1 / 2 d x d y , S δ ± ( j ) ( x , y , z ) = 0 d 1 0 d 2 Φ ± [ x - x , y - y , z - δ f ( x , y ) ] j ( x , y ) × { 1 + [ δ f x ( x , y ) ] 2 + [ δ f y ( x , y ) ] 2 } 1 / 2 d x d y , T δ ± ( j ) ( x , y , z ) = 0 d 1 0 d 2 Φ ± [ x - x , y - y , z - δ f ( x , y ) ] Div ( j ) ( x , y ) × { 1 + [ δ f x ( x , y ) ] 2 + [ δ f y ( x , y ) ] 2 } 1 / 2 d x d y .
j · n δ = j · n δ = 0 ,
Φ ± ( x , y , z ) = i 2 d 1 d 2 r = - s = - 1 γ r , s ± × exp ( i α r x + i β s y + i γ r , s ± z ) .
i 2 d 2 s = - 1 γ r , s ± exp ( i β s y + i γ r , s ± z ) = i 4 s = - exp ( - i s β d 2 ) H 0 ( 1 ) ( [ ( k ± ) 2 - α r 2 ] 1 / 2 ( y + s d 2 , z ) )
- μ 0 j - μ 0 n δ × [ D δ ± ( j ) - D δ - ( j ) ] + i ω μ 0 × n δ [ + S δ + ( j ) - - S δ - ( j ) ] - 1 i ω n δ × R δ ( j ) = - n δ × E i             on [ 0 , d 1 ] × [ 0 , d 2 ] , ( + + - ) 2 j + n δ × [ + D δ + ( j ) - - D δ - ( j ) ] + i ω μ 0 n δ × [ + S δ + ( j ) - - S δ - ( j ) ] - 1 i ω n δ × R δ ( j ) = - n δ × H i             on [ 0 , d 1 ] × [ 0 , d 2 ] ,
D δ ± ( j ) ( x , y ) = D δ ± ( j ) [ x , y , δ f ( x , y ) ] , S δ ± ( j ) ( x , y ) = S δ ± ( j ) [ x , y , δ f ( x , y ) ] , R δ ( j ) ( x , y ) = 0 d 1 0 d 2 [ j ( x , y ) · x , y ] · x , y [ Φ + ( x - x , y - y , δ f ( x , y ) - δ f ( x , y ) ] - Φ - [ x - x , y - y , δ f ( x , y ) - δ f ( x , y ) ] × { 1 + [ δ f x ( x , y ) ] 2 + [ δ f y ( x , y ) ] 2 } 1 / 2 d x d y .
F U Div ( V ) + F V · Grad ( U ) = 0 ,
U = Grad ( U ) + n δ U n δ ;
A δ = [ n δ × ( D δ + - D δ - ) - i ω n δ × ( + S δ + - - S δ - ) + 1 i ω μ 0 n δ × R δ 2 + + - [ i ω μ 0 n δ × ( + S δ + - - S δ - ) - 1 i ω μ 0 n δ × R δ ] 2 + + - n δ × ( + D δ + - - D δ - ) ] ,
( I + A δ ) ( J ) ( x , y ) = F δ ( x , y )
J ( x , y ) = [ j ( x , y ) j ( x , y ) ] , F δ ( x , y ) = [ 1 μ 0 n δ × E i [ x , y , δ f ( x , y ) ] - 2 + + - n δ × H i [ x , y , δ f ( x , y ) ] ] .
n δ × ( Φ ± × j ) ( x - x , y - y , δ f ( x , y ) - δ f ( x , y ) ) = { [ n δ ( x , y ) - n δ ( x , y ) ] · j } Φ ± - Φ ± n δ ( x , y ) j .
J , ν ( δ 0 ) = { J ( x , y ; δ ) = [ j ( x , y ; δ ) j ( x , y ; δ ) ] : J ( · ; δ ) is ( α , β ) - quasi - periodic , analytic for Im ( x ) < ν , Im ( y ) < ν , δ - δ 0 < and continuous for Im ( x ) ν , Im ( y ) ν , δ - δ 0 }
J J = sup { J ( x , y ; δ ) : Im ( x ) ν , Im ( y ) ν , δ - δ 0 } .
d k , ( r , s ) ± = [ d k , ( r , s ) 1 , ± d k , ( r , s ) 2 , ± d k , ( r , s ) 3 , ± ]
B r , s ± ( δ ) = k = 0 d k , ( r , s ) ± δ k .
H = 1 i ω μ 0 ( × E ) .
n δ × ( E + - E - ) = - ( n δ × A ) exp { i [ α x + β y - γ δ f ( x , y ) ] } , n δ × ( H + - H - ) = - ( n δ × B ) exp { i [ α x + β y - γ δ f ( x , y ) ] } on z = δ f ( x , y )
· E ± = 0             in ± z > ± δ f ( x , y )
n δ × ( E + - E - ) = - ( n δ × A ) exp { i [ α x + β y - γ δ f ( x , y ) ] } , n δ × ( × E + - × E - ) = - i ω μ 0 ( n δ × B ) × exp { i [ α x + β y - γ δ f ( x , y ) ] } on z = δ f ( x , y ) .
n δ = n δ ( x , y ) = 1 [ 1 + δ 2 f x ( x , y ) 2 + δ 2 f y ( x , y ) 2 ] 1 / 2 × ( - δ f x ( x , y ) , - δ f y ( x , y ) , 1 )
( - δ f x ( x , y ) , - δ f y ( x , y ) , 1 ) × { E + [ x , y , δ f ( x , y ) ; δ ] - E - [ x , y , δ f ( x , y ) ; δ ] } = - { ( - δ f x ( x , y ) , - δ f y ( x , y ) , 1 ) × A } × exp { i [ α x + β y - γ δ f ( x , y ) ] } ( - δ f x ( x , y ) , - δ f y ( x , y ) , 1 ) × { × E + [ x , y , δ f ( x , y ) ; δ ] - × E - [ x , y , δ f ( x , y ) ; δ ] } = - i { ( - δ f x ( x , y ) , - δ f y ( x , y ) , 1 ) × ( k × A ) } × exp { i [ α x + β y - γ δ f ( x , y ) ] } .
E 2 , + - E 2 , - + δ f y ( E 3 , + - E 3 , - ) = - ( A 2 + δ f y A 3 ) exp { i [ α x + β y - γ δ f ( x , y ) ] } , E 1 , + - E 1 , - + δ f x ( E 3 , + - E 3 , - ) = - ( A 1 + δ f x A 3 ) exp { i [ α x + β y - γ δ f ( x , y ) ] } , ( E z 1 , + - E z 1 , - ) - ( E x 3 , + - E x 3 , - ) + δ f y [ ( E x 2 , + - E x 2 , - ) - ( E y 1 , + - E y 1 , - ) ] = - i [ - γ A 1 - α A 3 + δ f y ( α A 2 - β A 1 ) ] × exp { i [ α x + β y - γ δ f ( x , y ) ] } , ( E y 3 , + - E y 3 , - ) - ( E z 2 , + - E z 2 , - ) + δ f x [ ( E x 2 , + - E x 2 , - ) - ( E y 1 , + - E y 1 , - ) ] = - i [ β A 3 + γ A 2 + δ f x ( α A 2 - β A 1 ) ] × exp { i [ α x + β y - γ δ f ( x , y ) ] } .
E x 1 , + + E y 2 , + + E z 3 , + = 0 , E x 1 , - + E y 2 , - + E z 3 , - = 0
E + = [ E 1 , + E 2 , + E 3 , + ] ,             E - = [ E 1 , - E 2 , - E 3 , - ]
1 n ! n ( E 2 , + - E 2 , - ) δ n = - k = 0 n - 1 f n - k ( n - k ) ! n - k z n - k [ 1 k ! k ( E 2 , + - E 2 , - ) δ k ] - f y k = 0 n - 1 f n - 1 - k ( n - 1 - k ) ! n - 1 - k z n - 1 - k × [ 1 k ! k ( E 3 , + - E 3 , - ) δ k ] - [ A 2 ( - i γ ) n f n n ! + A 3 ( - i γ ) n - 1 f y f n - 1 ( n - 1 ) ! ] × exp [ i ( α x + β y ) ] ; 1 n ! n ( E 1 , + - E 1 , - ) δ n = - k = 0 n - 1 f n - k ( n - k ) ! n - k z n - k [ 1 k ! k ( E 1 , + - E 1 , - ) δ k ] - f x k = 0 n - 1 f n - 1 - k ( n - 1 - k ) ! n - 1 - k z n - 1 - k × [ 1 k ! k ( E 3 , + - E 3 , - ) δ k ] - [ A 1 ( - i γ ) n f n n ! + A 3 ( - i γ ) n - 1 f x f n - 1 ( n - 1 ) ! ] × exp [ i ( α x + β y ) ] ; 1 n ! n δ n [ ( E z 1 , + - E z 1 , - ) - ( E x 3 , + - E x 3 , - ) ] = - k = 0 n - 1 f n - k ( n - k ) ! n - k z n - k { 1 k ! k δ k [ ( E z 1 , + - E z 1 , - ) - ( E x 3 , + - E x 3 , - ) ] } - f y k = 0 n - 1 f n - 1 - k ( n - 1 - k ) ! n - 1 - k z n - 1 - k { 1 k ! k δ k [ ( E x 2 , + - E x 2 , - ) - ( E y 1 , + - E y 1 , - ) ] ] } - i [ - ( γ A 1 + α A 3 ) ( - i γ ) n f n n ! + ( α A 2 - β A 1 ) ( - i γ ) n - 1 f y f n - 1 ( n - 1 ) ! ] × exp [ i ( α x + β y ) ] ; 1 n ! n δ n [ ( E y 3 , + - E y 3 , - ) - ( E z 2 , + - E z 2 , - ) ] = - k = 0 n - 1 f n - k ( n - k ) ! n - k z n - k { 1 k ! k δ k [ ( E y 3 , + - E y 3 , - ) - ( E z 2 , + - E z 2 , - ) ] } - f x k = 0 n - 1 f n - 1 - k ( n - 1 - k ) ! n - 1 - k z n - 1 - k { 1 k ! k δ k [ ( E x 2 , + - E x 2 , - ) - ( E y 1 , + - E y 1 , - ) ] } - i [ ( β A 3 + γ A 2 ) ( - i γ ) n f n n ! + ( α A 2 - β A 1 ) ( - i γ ) n - 1 f x f n - 1 ( n - 1 ) ! ] × exp [ i ( α x + β y ) ] ,
n E x 1 , + δ n + n E y 2 , + δ n + n E z 3 , + δ n = 0 , n E x 1 , - δ n + n E y 2 , - δ n + n E z 3 , - δ n = 0.
1 k ! k E ± δ k ( x , y , z ; 0 ) = r , s 1 k ! d k B r , s ± d δ k ( 0 ) exp ( i α r x + i β s y ± i γ r , s ± z ) = r , s d k , ( r , s ) ± exp ( i α r x + i β s y ± i γ r , s ± z ) .
f ( x , y ) l l ! = p = - l F l F q = - l F l F C l , ( p , q ) exp [ i ( K 1 p x + K 2 q y ) ] .
f x f l - 1 ( l - 1 ) ! = - l F p , q l F C l , ( p , q ) ( i K 1 p ) exp [ i ( K 1 p x + K 2 q y ) ] , f y f l - 1 ( l - 1 ) ! = - l F p , q l F C l , ( p , q ) ( i K 2 q ) exp [ i ( K 1 p x + K 2 q y ) ] ,
r , s [ d n , ( r , s ) 2 , + - d n , ( r , s ) 2 , - ] exp [ i ( α r x + β s y ) ] = - k = 0 n - 1 { - ( n - k ) F p , q ( n - k ) F C n - k , ( p , q ) exp [ i ( K 1 p x + K 2 q y ) ] } × { l , m [ ( i γ l , m + ) n - k d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k d k , ( l , m ) 2 , - ] exp [ i ( α l x + β m y ) ] } - k = 0 n - 1 { - ( n - k ) F p , q ( n - k ) F C n - k , ( p , q ) ( i K 2 q ) exp [ i ( K 1 p x + K 2 q y ) ] } × { l , m [ ( i γ l , m + ) n - 1 - k d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - 1 - k d k , ( l , m ) 3 , - ] exp [ i ( α l x + β m y ) ] } - ( A 2 ( - i γ ) n { - n F p , q n F C n , ( p , q ) exp [ i ( K 1 p x + K 2 q y ) ] } + A 3 ( - i γ ) n - 1 { - n F p , q n F C n , ( p , q ) ( i K 2 q ) exp [ i ( K 1 p x + K 2 q y ) ] } ) × exp [ i ( α x + β y ) ] ,
r , s { [ i γ r , s + d n , ( r , s ) 1 , + + i γ r , s - d n , ( r , s ) 1 , - ] - [ i α r d n , ( r , s ) 3 , + - i α r d n , ( r , s ) 3 , - ] } exp [ i ( α r x + β s y ) ] = - k = 0 n - 1 { - ( n - k ) F p , q ( n - k ) F C n - k , ( p , q ) exp [ i ( K 1 p x + K 2 q y ) ] } { l , m [ ( i γ l , m + ) n - k + 1 d k , ( l , m ) 1 , + - ( - i γ l . m - ) n - k + 1 d k , ( l , m ) 1 , - - ( i γ l , m + ) n - k ( i α r ) d k , ( l , m ) 3 , + + ( - i γ l , m - ) n - k ( i α r ) d k , ( l , m ) 3 , - ] × exp [ i ( α l x + β m y ) ] } - k = 0 n - 1 { - ( n - k ) F p , q ( n - k ) F C n - k , ( p , q ) ( i K 2 q ) exp [ i ( K 1 p x + K 2 q y ) ] } × { l , m [ ( i γ l , m + ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , - - ( i γ l , m + ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , + + ( - i γ l , m - ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , - ] exp [ i ( α l x + β m y ) ] } - i ( - ( γ A 1 + α A 3 ) ( - i γ ) n { - n F p , q n F C n , ( p , q ) exp [ i ( K 1 p x + K 2 q y ) ] } + ( α A 2 - β A 1 ) ( - i γ ) n - 1 { - n F p , q n F C n , ( p , q ) ( i K 2 q ) exp [ i ( K 1 p x + K 2 q y ) ] } ) × exp [ i ( α x + β y ) ] .
- - n F r , s n F [ A 2 ( - i γ ) n + A 3 ( - i γ ) n - 1 ( i K 2 s ) ] × C n , ( r , s ) exp [ i ( α r x + β s y ) ] - k = 0 n - 1 l , m - ( n - k ) F p , q ( n - k ) F [ ( i γ l , m + ) n - k d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k d k , ( l , m ) 2 , - ] C n - k , ( p , q ) × exp [ i ( K 1 p x + K 2 q y ) ] exp [ i ( α l x + β m y ) ] - k = 0 n - 1 l , m - n ( n - k ) F p , q ( n - k ) F [ ( i γ l , m + ) n - 1 - k d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - 1 - k d k , ( l , m ) 3 , - ] ( i K 2 q ) C n - k , ( p , q ) × exp [ i ( K 1 p x + K 2 q y ) ] exp [ i ( α l x + β m y ) ] ,
- i - n F r , s n F [ - ( γ A 1 + α A 2 ) ( - i γ ) n + ( α A 2 - β A 1 ) ( - i γ ) n - 1 ( i K 2 s ) ] C n , ( r , s ) exp [ i ( α r x + β s y ) ] - k = 0 n - 1 l , m - ( n - k ) F p , q ( n - k ) F [ ( i γ l , m + ) n - k - 1 d k , ( l , m ) 1 , + - ( - i γ l , m - ) n - k + 1 d k , ( l , m ) 1 , - - ( i γ l , m + ) n - k ( i α r ) d k , ( l , m ) 3 , + + ( - i γ l , m - ) n - k ( i α r ) d k , ( l , m ) 3 , - ] C n - k , ( p , q ) × exp [ i ( K 1 p x + K 2 q y ) ] exp [ i ( α l x + β m y ) ] - k = 0 n - 1 l , m - ( n - k ) F p , q ( n - k ) F [ ( i γ l , m + ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , - - ( i γ l , m + ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , + + ( - i γ l , m - ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , - ] ( i K 2 q ) C n - k , ( p , q ) × exp [ i ( K 1 p x + K 2 q y ) ] exp [ i ( α l x + β m y ) ] .
exp [ i ( K 1 p x + K 2 q y ) ] exp [ i ( α l x + β m y ) ] = exp [ i ( α p + l x + β q + m y ) ] ,
- - n F r , s n F [ A 2 ( - i γ ) n + A 3 ( - i γ ) n - 1 ( i K 2 s ) ] C n , ( r , s ) exp [ i ( α r x + β s y ) ] - k = 0 n - 1 l , m l - ( n - k ) F r l + ( n - k ) F m - ( n - k ) F s m + ( n - k ) F { [ ( i γ l , m + ) n - k d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k d k , ( l , m ) 2 , - ] + [ ( i γ l , m + ) n - 1 - k d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - 1 - k d k , ( l , m ) 3 , - ] [ i K 2 ( s - m ) ] } C n - k , ( r - l , s - m ) exp [ i ( α r x + β s y ) ] , - i - n F r , s n F [ - ( γ A 1 + α A 3 ) ( - i γ ) n + ( α A 2 - β A 1 ) ( - i γ ) n - 1 ( i K 2 s ) ] C n , ( r , s ) exp [ i ( α r x + β s y ) ] - k = 0 n - 1 l , m l - ( n - k ) F r l + ( n - k ) F m - ( n - k ) F s m + ( n - k ) F { [ ( i γ l , m + ) n - k + 1 d k , ( l , m ) 1 , + - ( - i γ l , m - ) n - k + 1 d k , ( l , m ) 1 , - - ( i γ l , m + ) n - k ( i α r ) d k , ( l , m ) 3 , + + ( - i γ l , m - ) n - k ( i α r ) d k , ( l , m ) 3 , - ] + [ ( i γ l , m + ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k - 1 ( i α r ) d k , ( l , m ) 2 , - - ( i γ l , m + ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , + + ( - i γ l , m - ) n - k - 1 ( i β s ) d k , ( l , m ) 1 , - ] [ ( i K 2 ( s - m ) ] } C n - k , ( r - l , s - m ) exp [ i ( α r x + β s y ) ] .
d k , ( l , m ) i , ± = 0             if l > k F             or m > k F .
d n , ( r , s ) 2 , + - d n , ( r , s ) 2 , + = - [ A 2 ( - i γ ) n + A 3 ( - i γ ) n - 1 ( i K 2 s ) ] C n , ( r , s ) - k = 0 n - 1 l = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] m = max [ - k F , s - ( n - k ) F ] min [ k F , s + ( n - k ) F ] { ( i γ l , m + ) n - k d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k d k , ( l , m ) 2 , - + [ ( i γ l , m + ) n - 1 - k d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - 1 - k d k , ( l , m ) 3 , - ] [ i K 2 ( s - m ) ] } C n - k , ( r - l , s - m ) ,
γ r , s + d n , ( r , s ) 1 , + + γ r , s - d n , ( r , s ) 1 , - - α r d n , ( r , s ) 3 , + + α r d n , ( r , s ) 3 , - = - [ - ( γ A 1 + α A 3 ) ( - i γ ) n + ( α A 2 - β A 1 ) ( - i γ ) n - 1 ( i K 2 s ) ] C n , ( r , s ) - k = 0 n - 1 l = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] m = max [ - k F , s - ( n - k ) F ] min [ k F , s + ( n - k ) F ] { γ l , m + ( i γ l , m + ) n - k d k , ( l , m ) 1 , + + γ l , m - ( - i γ l , m - ) n - k d k , ( l , m ) 1 , - - ( i γ l , m + ) n - k α r d k , ( l , m ) 3 , + + ( - i γ l , m - ) n - k α r d k , ( l , m ) 3 , - + [ ( i γ l , m + ) n - k - 1 α r d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k - 1 α r d k , ( l , m ) 2 , - - ( i γ l , m + ) n - k - 1 β s d k , ( l , m ) 1 , + + ( - i γ l , m - ) n - k - 1 β s d k , ( l , m ) 1 , - ] [ i K 2 ( s - m ) ] } C n - k , ( r - l , s - m ) .
d n , ( r , s ) 1 , + - d n , ( r , s ) 1 , - = - [ A 2 ( - i γ ) n + A 3 ( - i γ ) n - 1 ( i K 1 r ) ] C n , ( r , s ) - k = 0 n - 1 l = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] m = max [ - k F , s - ( n - k ) F ] min [ k F , s + ( n - k ) F ] { ( i γ l , m + ) n - k d k , ( l , m ) 1 , + - ( - i γ l , m - ) n - k d k , ( l , m ) 1 , - + [ ( i γ l , m + ) n - 1 - k d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - 1 - k d k , ( l - m ) 3 , - ] [ i K 1 ( r - l ) ] } C n - k , ( r - l , s - m ) ,
β r d n , ( r , s ) 3 , + - β r d n , ( r , s ) 3 , - - γ r , s + d n , ( r , s ) 2 , + - γ r , s - d n , ( r , s ) 2 , - = - [ ( γ A 2 + β A 2 ) ( - i γ ) n + ( α A 2 - β A 1 ) ( - i γ ) n - 1 ( i K 1 r ) ] C n , ( r , s ) - k = 0 n - 1 l = max [ - k F , r - ( n - k ) F ] min [ k F , r + ( n - k ) F ] m = max [ - k F , s - ( n - k ) F ] min [ k F , s + ( n - k ) F ] { ( i γ l , m + ) n - k β s d k , ( l , m ) 3 , + - ( - i γ l , m - ) n - k β s d k , ( l , m ) 3 , - - γ l , m + ( i γ l , m + ) n - k d k , ( l , m ) 2 , + - γ l , m - ( - i γ l , m - ) n - k d k , ( l , m ) 2 , - + [ ( i γ l , m + ) n - k - 1 α r d k , ( l , m ) 2 , + - ( - i γ l , m - ) n - k - 1 α r d k , ( l , m ) 2 , - - ( i γ l , m + ) n - k - 1 β s d k , ( l , m ) 1 , + + ( - i γ l , m - ) n - k - 1 β s d k , ( l , m ) 1 , - ] [ i K 1 ( r - l ) ] } C n - k , ( r - l , s - m ) .
α r d n , ( r , s ) 1 , + + β s d n , ( r , s ) 2 , + + γ r , s + d n , ( r , s ) 3 , + = 0 ,
α r d n , ( r , s ) 1 , - + β s d n , ( r , s ) 2 , - + γ r , s - d n , ( r , s ) 3 , - = 0.
d k , ( l , m ) i , - = 0             for i = 1 , 2 , 3             and all k , l , and m .
f ( x , y ) = h 4 [ cos ( 2 π x d ) + cos ( 2 π y d ) ] ,
= 1 - [ ( r , s ) U + e r , s + + ( r , s ) U - e r , s - ] .

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