Abstract

Previously ( Proc. R. Soc. Edinburgh 122A, 317– 340, 1992) we established that solutions to problems of diffraction of light in a periodic structure behave analytically with respect to variations of the interface. We present an algorithm based on this observation for the numerical solution of the problem. The principal component of the algorithm is a simple recursive formula for the derivatives of the efficiencies with respect to the height of the grating. A conformal mapping mechanism is introduced to enhance the convergence of the series. This allows us to deal with the types of profile and wavelength usually considered in practice. To illustrate our method, we give numerical results for sinusoidal and echelette gratings.

© 1993 Optical Society of America

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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 122A, 317–340 (1992).
    [CrossRef]
  2. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [CrossRef]
  3. R. F. Millar, “The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers,” Radio Sci. 8, 785–796 (1973).
    [CrossRef]
  4. W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rational Mech. Anal. 5, 323–334 (1956).
  5. D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
    [CrossRef]
  6. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
    [CrossRef]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. R. Petit, M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinitement conducteur,”C. R. Acad. Sci. Ser. B 262, 468–471 (1966).
  9. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).,
    [CrossRef]
  10. R. Petit, “Étude numérique de la diffraction par un réseau,”C. R. Acad. Sci. Paris 260, 4454–4457 (1965).
  11. R. Petit, “Étude numérique de la diffraction par un réseau métallique,”C. R. Acad. Sci. Paris 261, 4677–4680 (1965).
  12. J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
    [CrossRef]
  13. P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971).
  14. J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansion used to describe the field diffracted by a grating,”J. Opt. Soc. Am. 71, 593–598 (1981).
    [CrossRef]
  15. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
    [CrossRef]
  16. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
  17. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
  18. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),”J. Opt. Soc. Am. 41, 213–222 (1941).
    [CrossRef]
  19. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 48–52.
  20. G. A. Baker, P. Graves-Morris, Padé Approximants. Part I: Basic Theory (Addison-Wesley, Massachusetts, 1981).
  21. O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” submitted to Math. Comp.
  22. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Dielectric gratings, Padé approximants and singularities,” submitted to J. Opt. Soc. Am. A.

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 122A, 317–340 (1992).
[CrossRef]

1981 (1)

1973 (1)

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

1971 (2)

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

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

1970 (1)

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

1966 (1)

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

1965 (3)

R. Petit, “Étude numérique de la diffraction par un réseau,”C. R. Acad. Sci. Paris 260, 4454–4457 (1965).

R. Petit, “Étude numérique de la diffraction par un réseau métallique,”C. R. Acad. Sci. Paris 261, 4677–4680 (1965).

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

1956 (1)

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

1941 (1)

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),”J. Opt. Soc. Am. 41, 213–222 (1941).
[CrossRef]

1907 (2)

Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

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

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Baker, G. A.

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

Bousquet, J.

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

Bruno, O. P.

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]

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” submitted to Math. Comp.

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Dielectric gratings, Padé approximants and singularities,” submitted to J. Opt. Soc. Am. A.

Cadilhac, M.

J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansion used to describe the field diffracted by a grating,”J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

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

Fano, U.

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),”J. Opt. Soc. Am. 41, 213–222 (1941).
[CrossRef]

Graves-Morris, P.

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

Hugonin, J. P.

Maystre, D.

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

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
[CrossRef]

Meecham, W. C.

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

Millar, R. F.

R. F. Millar, “The Rayleigh hypothesis and a related least-squares 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 II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).,
[CrossRef]

Nevière, M.

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
[CrossRef]

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 48–52.

Pavageau, J.

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

Petit, R.

J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansion used to describe the field diffracted by a grating,”J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

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

R. Petit, “Étude numérique de la diffraction par un réseau,”C. R. Acad. Sci. Paris 260, 4454–4457 (1965).

R. Petit, “Étude numérique de la diffraction par un réseau métallique,”C. R. Acad. Sci. Paris 261, 4677–4680 (1965).

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
[CrossRef]

Rayleigh,

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

Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

Reitich, F.

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]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Dielectric gratings, Padé approximants and singularities,” submitted to J. Opt. Soc. Am. A.

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” submitted to Math. Comp.

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).

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

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).

C. R. Acad. Sci. Paris (2)

R. Petit, “Étude numérique de la diffraction par un réseau,”C. R. Acad. Sci. Paris 260, 4454–4457 (1965).

R. Petit, “Étude numérique de la diffraction par un réseau métallique,”C. R. Acad. Sci. Paris 261, 4677–4680 (1965).

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

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

J. Opt. Soc. Am. (2)

J. P. Hugonin, R. Petit, M. Cadilhac, “Plane-wave expansion used to describe the field diffracted by a grating,”J. Opt. Soc. Am. 71, 593–598 (1981).
[CrossRef]

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),”J. Opt. Soc. Am. 41, 213–222 (1941).
[CrossRef]

J. Rational Mech. Anal. (1)

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

Opt. Acta (1)

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

Philos. Mag. (2)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).

Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

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 (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]

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

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

Radio Sci. (1)

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

Other (7)

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980).
[CrossRef]

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

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

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 48–52.

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

O. P. Bruno, F. Reitich, “Approximation of analytic functions: a method of enhanced convergence,” submitted to Math. Comp.

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. II. Dielectric gratings, Padé approximants and singularities,” submitted to J. Opt. Soc. Am. A.

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Figures (2)

Fig. 1
Fig. 1

Circles of convergence D1 and D2 of the functions g(w) and g(ζ) about w = 1 and ζ = 1, respectively.

Fig. 2
Fig. 2

The region C of analyticity of the Rayleigh coefficients Br(δ) and the lens-shaped region L that is conformally transformed onto the right half-plane by means of g(δ) = [(Aδ)/(A + δ)]α.

Tables (7)

Tables Icon

Table 1 Comparison between Direct and Enhanced Convergence: First-Order Efficiency for a Sinusoidal Grating in TE Polarization, h/d = 0.3

Tables Icon

Table 2 Comparison between Direct and Enhanced Convergence: First-Order Efficiency for a Sinusoidal Grating in TE Polarization, h/d = 0.4

Tables Icon

Table 3 Efficiencies for f(x) = h/2 cos(2πx/d) in TE Polarization with A = 9, σ = 0.13, and 60 Terms of the Taylor Expansion

Tables Icon

Table 4 Efficiencies for f(x) = h/2 cos(2πx/d) in TM Polarization with A = 9, σ = 0.13, and 60 Terms of the Taylor Expansion

Tables Icon

Table 5 Efficiencies for f(x) = h/2g(2πx/d) in TE Polarization with A = 9, σ = 0.20, F = 10, and 30 Terms of the Taylor Expansion

Tables Icon

Table 6 Efficiencies for f(x) = h/2g(2πx/d) in TM Polarization with A = 9, σ = 0.20, F = 10, and 40 Terms of the Taylor Expansion

Tables Icon

Table 7 Efficiencies in Quadruple Precision Arithmetic for f(x) = h/2 cos(2πx/d) in TE Polarization with A = 9, σ = 0.13, and 100 Terms of the Taylor Expansion

Equations (55)

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y = ( h / 2 ) sin ( 2 π x / d ) ,
y = δ f ( x )
y = 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 ) ,
E up = E i + E ref 1 , H up = H i + H ref 1 ,
× E = i ω μ 0 H , · E = 0 , × H = i ω + E , · H = 0
n × E up = 0             on             y = f ( x ) ,
n × ( E up - E down ) = 0 ,             n × ( H up - H down ) = 0             on             y = f ( x ) .
v ( x + d , y ) = exp ( i α d ) v ( x , y ) .
Δ u + ( k ± ) 2 u = 0             in             Ω ± ,
u = - exp [ i α x - i β f ( x ) ]             on             y = f ( x ) .
u n = - n exp ( i α x - i β y )             on             y = f ( x ) .
u + - u - = - exp [ i α x - i β f ( x ) ] on y = f ( x ) , u + n - u - n = - n exp ( i α x - i β y ) on y = f ( x ) .
u + - u - = - exp [ i α x - i β f ( x ) ] on y = f ( x ) , u + n - 1 ν 0 2 u - n = - n exp ( i α x - i β y ) on y = f ( x ) ,
ν 0 2 = - + = ( k - k + ) 2 .
K = 2 π d ,             α n = α + n K ,             α n 2 + ( β n ± ) 2 = ( k ± ) 2 ,
k + ± ( α + n K ) ,             k - ± ( α + n K ) ,
u + = n = - A n + exp ( i α n x - i β n + y ) + n = - B n + exp ( i α n x + i β n + y ) .
u - = n = - A n - exp ( i α n x - i β n - y ) + n = - B n - exp ( i α n x + i β n - y ) .
A n + = 0 ,             B n - = 0             for all n .
n U + β n + B n + 2 = β 0 + ,
U + { n : β n + > 0 } .
n U + e n = 1 ,
u ( x , y , δ ) = r = - B r ( δ ) exp ( i α r x + i β r y ) .
f ( x ) = r = - F F C 1 , r exp ( i K r x ) ,             K = 2 π d .
f δ ( x ) = δ f ( x ) ,
u [ x , δ f ( x ) , δ ] = - u i [ x , δ f ( x ) ] = - exp [ i α x - i β δ f ( x ) ] .
B r ( δ ) = k = 0 d k , r δ k ,
1 k ! k u δ k ( x , y , 0 ) = r 1 k ! d k B r d δ k ( 0 ) exp ( i α r x + i β r y ) ,
d k , r = 1 k ! d k B r d δ k ( 0 ) ,
1 k ! k u δ k ( x , y , 0 ) = r d k , r exp ( i α r x + i β r y ) .
1 n ! n u δ n ( x , 0 , 0 ) = - ( - i β ) n n ! f ( x ) n exp ( i α x ) - k = 0 n - 1 f ( x ) n - k ( n - k ) ! n - k y n - k ( 1 k ! k u δ k ) ( x , 0 , 0 ) .
1 l ! f ( x ) = r = - l F l F C l , r exp ( i K r x ) .
r = - d n , r exp ( i α r x ) = - [ r = - n F n F ( - i β ) n C n , r exp ( i K r x ) ] exp ( i α x ) - k = 0 n - 1 [ p = - ( n - k ) F ( n - k ) F C n - k , p exp ( i K p x ) ] × n - k y n - k [ q = - d k , q exp ( i α q x + i β q y ) ] | y = 0 ,
r = - d n , 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 - l ) F ( n - k ) F C n - k , p ( i β q ) n - k × d k , q exp ( i K p x + i α q x ) .
exp ( i K p x + i α q x ) = exp ( i α p + q x ) ,
r = - d n , 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 = - r = q - ( n - k ) F q + ( n - k ) F C n - k , r - 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 ] exp ( i α r x ) .
d k , q = 0             if             q > k F .
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 ,
d n , r = - i β r ( - i β ) n - 1 C n , r ( β 2 - α r K ) - i β 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 - 1 × [ ( β q ) 2 - α q ( r - q ) K ] d k , q .
f ( x ) = 2 cos ( K x ) = exp ( i K x ) + exp ( - i K x ) ,
C n , k = 1 n ! ( ( n - k n ) / 2 )
d 0 , 0 = - 1 d 1 , - 1 d 1 , 1 d 2 , - 2 d 2 , 0 d 2 , 2 d 3 , - 3 d 3 , - 1 d 3 , 1 d 3 , 3 d 4 , - 4 d 4 , - 2 d 4 , 0 d 4 , 2 d 4 , 4
d 0 , 0 = - 1 d 1 , - 1 d 1 , 1 d 2 , - 2 d 2 , 0 d 2 , 2 d 3 , - 1 d 3 , 1 d 4 , 0
f ( x ) = 2 cos ( K x ) + ( 1 / 5 ) cos ( 3 K x ) ,
{ arg ( w ) < π / ( 2 α ) }
ζ = w α
g [ w ( ζ ) ] = g ( ζ 1 / α )
w = A - δ A + δ .
π / ( 2 α ) = arctan [ 2 A σ / ( A 2 - σ 2 ) ] .
= 1 - n U + e n .
y = ( h / 2 ) cos ( 2 π x / d ) ,
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 ) ,

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