Abstract

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green’s function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A 26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green’s functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R.Petit, ed., Electromagnetic Theory of Gratings (Speinger-Verlag, 1980).
    [CrossRef]
  2. G.Bao, L.Cowsar, and W.Masters, eds., Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  3. M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).
  4. G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]
  5. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  6. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  7. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  8. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible, et l’ultraviolet,” (Ph.D. dissertation, Université d’Aix-Marseille III, France, 1974).
  9. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), Chapter 3.
  10. A. Pomp, “Integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
    [CrossRef]
  11. B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
    [CrossRef]
  12. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  13. E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
    [CrossRef]
  14. T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418(2005).
    [CrossRef]
  15. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).
  16. L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597(2010).
    [CrossRef]
  17. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  18. I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
    [CrossRef]
  19. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
    [CrossRef]
  20. K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
    [CrossRef]
  21. Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A 26, 2444–2451 (2009).
    [CrossRef]
  22. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442–1449 (2009).
    [CrossRef]
  23. Y. Y. Lu and J. R. McLaughlin, “Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446(1996).
    [CrossRef]
  24. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. 24, 3448–3453 (2006).
    [CrossRef]
  25. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466–1473 (2008).
    [CrossRef]
  26. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  27. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).
  28. L. Wang, J. A. Cox, and A. Friedman, “Modal analysis of homogeneous optical waveguides by the boundary integral formulation and the Nyström method,” J. Opt. Soc. Am. A 15, 92–100 (1998).
    [CrossRef]
  29. S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
    [CrossRef]
  30. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  31. L. Li, J. Chandezon, G. Granet, and J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
    [CrossRef]
  32. L. Li, “Modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  33. M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photon. News 10, 44–48 (1999).
    [CrossRef]

2010 (3)

2009 (2)

2008 (1)

2007 (1)

2006 (2)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. 24, 3448–3453 (2006).
[CrossRef]

2005 (2)

2002 (1)

1999 (3)

1998 (1)

1997 (1)

1996 (6)

1993 (2)

L. Li, “Modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

1991 (1)

A. Pomp, “Integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

1982 (1)

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Bao, G.

Bozhkov, B.

Chandezon, J.

Chen, Z. M.

Chuang, S. L.

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Cox, J. A.

Friedman, A.

Goray, L. I.

Gralak, B.

Granet, G.

Guizal, B.

Gundu, K. M.

Gushchin, I.

Hoose, J.

Huang, Y.

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. 24, 3448–3453 (2006).
[CrossRef]

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Kong, J. A.

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Lalanne, P.

Li, L.

Lu, Y. Y.

Lyndin, N. M.

Mafi, A.

Magath, T.

Mait, J. N.

Mansuripur, M.

M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photon. News 10, 44–48 (1999).
[CrossRef]

Maystre, D.

E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
[CrossRef]

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible, et l’ultraviolet,” (Ph.D. dissertation, Université d’Aix-Marseille III, France, 1974).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), Chapter 3.

McLaughlin, J. R.

Y. Y. Lu and J. R. McLaughlin, “Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446(1996).
[CrossRef]

Mirotznik, M. S.

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Morris, G. M.

Nevière, M.

Parriaux, O.

Plumey, J. P.

Pomp, A.

A. Pomp, “Integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

Popov, E.

Prather, D. W.

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).

Schmidt, G.

L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597(2010).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).

Serebryannikov, A. E.

Tayeb, G.

Tishchenko, A. V.

Wang, L.

Wu, H. J.

Wu, Y.

Wyrowski, F.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Yeh, W.-H.

M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photon. News 10, 44–48 (1999).
[CrossRef]

Appl. Opt. (2)

CiCP (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “Fast integral equation method for diffraction gratings,” CiCP 1, 984–1009(2006).

J. Acoust. Soc. Am. (1)

Y. Y. Lu and J. R. McLaughlin, “Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446(1996).
[CrossRef]

J. Lightw. Technol. (1)

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightw. Technol. 24, 3448–3453 (2006).
[CrossRef]

J. Mod. Opt. (3)

L. Li, “Modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

A. Pomp, “Integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

J. Opt. Soc. Am. A (15)

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597(2010).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
[CrossRef]

I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
[CrossRef]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
[CrossRef]

K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
[CrossRef]

Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A 26, 2444–2451 (2009).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418(2005).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

L. Wang, J. A. Cox, and A. Friedman, “Modal analysis of homogeneous optical waveguides by the boundary integral formulation and the Nyström method,” J. Opt. Soc. Am. A 15, 92–100 (1998).
[CrossRef]

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

Opt. Photon. News (1)

M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photon. News 10, 44–48 (1999).
[CrossRef]

Radio Sci. (1)

S. L. Chuang and J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Other (6)

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible, et l’ultraviolet,” (Ph.D. dissertation, Université d’Aix-Marseille III, France, 1974).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), Chapter 3.

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

G.Bao, L.Cowsar, and W.Masters, eds., Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

M. Nevière and E. Popov, Light Propagation in Periodic Media (Marcel Dekker, 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Rectangular domain Σ for one period of a diffraction grating.

Fig. 2
Fig. 2

Dielectric sinusoidal diffraction grating.

Fig. 3
Fig. 3

Metallic lamellar diffraction grating.

Fig. 4
Fig. 4

Metallic echelette diffraction grating.

Tables (3)

Tables Icon

Table 1 Diffraction Efficiencies of Sinusoidal Grating in a Conical Mounting Obtained by C Method, BIE Method [16], and BIE-NtD Method

Tables Icon

Table 2 Diffraction Efficiencies of Metallic Lamellar Grating in a Conical Mounting Calculated Using FMM, BIE Method [16], and BIE-NtD Method

Tables Icon

Table 3 Diffraction Efficiencies of Metallic Echelette Grating in a Conical Mounting Obtained by C Method, BIE Method [16], and BIE-NtD Method

Equations (26)

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

x 2 u + y 2 u + η u = 0
ε η ν E z + γ 0 k 0 η τ H ˜ z , μ η ν H ˜ z γ 0 k 0 η τ E z
u ( i ) ( r ) = a 0 exp [ i ( α 0 x β 0 ( 1 ) y ) ] , y > D ,
u ( r ) ( r ) = j = b j exp [ i ( α j x + β j ( 1 ) y ) ] , y > D ,
u ( t ) ( r ) = j = c j exp [ i ( α j x β j ( 2 ) y ) ] , y < 0 ,
α j = α 0 + 2 π j / L , β j ( p ) = η ( p ) α j 2 , η ( p ) = k 0 2 ε ( p ) μ ( p ) γ 0 2 ,
u ( x + L , y ) = exp ( i α 0 L ) u ( x , y ) .
y u = B ( 2 ) u , y = 0 ,
y u = B ( 1 ) u 2 B ( 1 ) u ( i ) , y = D + .
B 0 ( 1 ) exp ( i α j x ) = i β j ( 1 ) exp ( i α j x ) , j = 0 , ± 1 , ± 2 ,
Q j + u j = ν u j + , Q j u j = ν u j , Y j u j = u 0 ,
( Q m + B ( 1 ) ) u m = 2 B ( 1 ) u ( i ) | y = D + ,
u ( r ) | y = D + = u m u ( i ) | y = D + , u ( t ) | y = 0 = u 0 = Y m u m .
N j [ ν u j 1 + ν u j ] = [ N j , 11 N j , 12 N j , 21 N j , 22 ] [ ν u j 1 + ν u j ] = [ u j 1 u j ] ,
Z = ( I [ N j , 11 N j , 11 ] Q j 1 + ) 1 [ N j , 12 N j , 12 ] ,
Q j = ( [ N j , 22 N j , 22 ] + [ N j , 21 N j , 21 ] Q j 1 + Z ) 1 ,
Y j = Y j 1 Z Q j .
Q j + = [ σ 1 0 0 σ 2 ] Q j + [ 0 σ 3 τ σ 4 τ 0 ] ,
σ 1 = ε η + ε + η , σ 2 = μ η + μ + η , σ 3 = γ 0 ( η + η ) k 0 η ε + , σ 4 = γ 0 ( η + η ) k 0 η μ + .
B ( 2 ) = S B ˜ ( 2 ) S ,
Q ˜ m + = S Q m + S , Y ˜ m = S Y m S ,
r = r ( t ) = ( x ( t ) , y ( t ) ) ,
τ u ( r ) = 1 | r ( t ) | d u ( r ( t ) ) d t ,
min c 0 , c 1 , , c n l = 1 q | k = 0 n c k T k ( t ˜ l ) u ( t l ) | 2 ,
u ( t l ) 2 t q + 1 t 0 k = 0 n c k T k ( t ˜ l ) .
τ [ u ( t 1 ) u ( t 2 ) u ( t q ) ] C [ u ( t 1 ) u ( t 2 ) u ( t q ) ] , C = C 0 C 2 C 1 ,

Metrics