Abstract

Off-plane scattering of time-harmonic plane waves by a plane diffraction grating with arbitrary conductivity and general surface profile is considered in a rigorous electromagnetic formulation. Integral equations for conical diffraction are obtained involving, besides the boundary integrals of the single and double layer potentials, singular integrals, the tangential derivative of single-layer potentials. We derive an explicit formula for the calculation of the absorption in conical diffraction. Some rules that are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive gratings, surfaces with edges, real profiles, and gratings working at short wavelengths.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  17. L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23, 155-165 (2006).
    [CrossRef]
  18. B. H. Kleemann and J. Erxmeyer, “Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004).
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    [CrossRef]
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  30. L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390-30, 73900V (2009).
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  35. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
    [CrossRef]
  36. E. Popov, B. Chernov, M. Neviere, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199-206 (2004).
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    [CrossRef] [PubMed]

2009 (1)

L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390-30, 73900V (2009).
[CrossRef]

2007 (1)

R. Köhle, “Rigorous simulation study of mask gratings at conical illumination,” Proc. SPIE 6607, 66072Z (2007).
[CrossRef]

2006 (4)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

L. I. Goray, J. F. Seely, and S. Yu. Sadov, “Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme ultraviolet gratings at near normal incidence,” J. Appl. Phys. 100, 094901 (2006).
[CrossRef]

L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23, 155-165 (2006).
[CrossRef]

J. F. Seely, L. I. Goray, B. Kjornrattanawanich, J. M. Laming, G. E. Holland, K. A. Flanagan, R. K. Heilmann, C.-H. Chang, M. L. Schattenburg, and A. P. Rasmussen, “Efficiency of a grazing-incidence off-plane grating in the soft-x-ray region,” Appl. Opt. 45, 1680-1687 (2006).
[CrossRef] [PubMed]

2004 (2)

E. Popov, B. Chernov, M. Neviere, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199-206 (2004).
[CrossRef]

B. H. Kleemann and J. Erxmeyer, “Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004).
[CrossRef]

2002 (2)

2001 (2)

2000 (1)

J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).

1999 (2)

1997 (2)

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]

B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997).
[CrossRef]

1996 (3)

1995 (1)

J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995).
[CrossRef]

1994 (1)

1993 (2)

L. Li, “A 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 (2)

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

1990 (1)

1986 (1)

E. Popov and L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175-180 (1986).
[CrossRef]

Ao, C. O.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001), pp. 61-110.

Bendickson, J. M.

Bonod, N.

Bozhkov, B.

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

Chandezon, J.

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247-2255 (1996).
[CrossRef]

J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995).
[CrossRef]

Chang, C.-H.

Chernov, B.

Content, D. A.

Ding, K.-H.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001), pp. 61-110.

Elschner, J.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

Erxmeyer, J.

B. H. Kleemann and J. Erxmeyer, “Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004).
[CrossRef]

Flanagan, K. A.

Gatzke, J.

B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Goray, L. I.

L. I. Goray, “Specular and diffuse scattering from random asperities of any profile using the rigorous method for x-rays and neutrons,” Proc. SPIE 7390-30, 73900V (2009).
[CrossRef]

J. F. Seely, L. I. Goray, B. Kjornrattanawanich, J. M. Laming, G. E. Holland, K. A. Flanagan, R. K. Heilmann, C.-H. Chang, M. L. Schattenburg, and A. P. Rasmussen, “Efficiency of a grazing-incidence off-plane grating in the soft-x-ray region,” Appl. Opt. 45, 1680-1687 (2006).
[CrossRef] [PubMed]

L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23, 155-165 (2006).
[CrossRef]

L. I. Goray, J. F. Seely, and S. Yu. Sadov, “Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme ultraviolet gratings at near normal incidence,” J. Appl. Phys. 100, 094901 (2006).
[CrossRef]

L. I. Goray and J. F. Seely, “Efficiencies of master replica, and multilayer gratings for the soft-x-ray-extreme-ultraviolet range: modeling based on the modified integral method and comparisons with measurements,” Appl. Opt. 41, 1434-1445 (2002).
[CrossRef] [PubMed]

L. I. Goray, “Modified integral method for weak convergence problems of light scattering on relief grating,” Proc. SPIE 4291, 1-12 (2001).
[CrossRef]

L. I. Goray and S. Yu. Sadov, “Numerical modeling of coated gratings in sensitive cases,” in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), 365-379.

Granet, G.

J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995).
[CrossRef]

Hafner, Ch.

Ch. Hafner, Post-modern Electromagnetics: Using Intelligent Maxwell Solvers (Wiley, 1999).

Heilmann, R. K.

Hinder, R.

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).

Holland, G. E.

Hoose, J.

Jung, Ch.

B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997).
[CrossRef]

Kjornrattanawanich, B.

Kleemann, B.

B. 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]

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

B. H. Kleemann and J. Erxmeyer, “Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004).
[CrossRef]

B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997).
[CrossRef]

Köhle, R.

R. Köhle, “Rigorous simulation study of mask gratings at conical illumination,” Proc. SPIE 6607, 66072Z (2007).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001), pp. 61-110.

Kuznetsov, I. G.

Laming, J. M.

Li, L.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Vol. 58 of Optical Engineering Series (Marcel Dekker, 1997), pp. 367-399.

Mait, J. N.

Mansuripur, M.

M. Mansuripur, L. Li, and W.-H. Yeh, “Diffraction gratings: part 2,” Opt. Photonics News 10, August 1, 1999, pp. 44-48.
[CrossRef]

Mashev, L.

E. Popov and L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175-180 (1986).
[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]

M. Saillard and D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982-990 (1990).
[CrossRef]

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

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R.Petit, ed. (Springer, 1980), pp. 53-100.
[CrossRef]

Mirotznik, M. S.

Mitreiter, A.

B. 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]

Nelles, B.

B. H. Kleemann, J. Gatzke, Ch. Jung, and B. Nelles, “Design and efficiency characterization of diffraction gratings for applications in synchrotron monochromators by electromagnetic methods and its comparison with measurement,” Proc. SPIE 3150, 137-147 (1997).
[CrossRef]

Neviere, M.

E. Popov, B. Chernov, M. Neviere, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199-206 (2004).
[CrossRef]

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).

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

Penzel, F.

J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).

Petit, R.

F. Zolla and R. Petit, “Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts,” J. Opt. Soc. Am. A 13, 796-802 (1996).
[CrossRef]

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

Plumey, J. P.

J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995).
[CrossRef]

Pomp, A.

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

Popov, E.

E. Popov, B. Chernov, M. Neviere, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199-206 (2004).
[CrossRef]

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]

E. Popov and L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. 17, 175-180 (1986).
[CrossRef]

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).

E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Vol. 58 of Optical Engineering Series (Marcel Dekker, 1997), pp. 367-399.

Prather, D. W.

Rasmussen, A. P.

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

Sadov, S. Yu.

L. I. Goray, I. G. Kuznetsov, S. Yu. Sadov, and D. A. Content, “Multilayer resonant subwavelength gratings: effects of waveguide modes and real groove profiles,” J. Opt. Soc. Am. A 23, 155-165 (2006).
[CrossRef]

L. I. Goray, J. F. Seely, and S. Yu. Sadov, “Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme ultraviolet gratings at near normal incidence,” J. Appl. Phys. 100, 094901 (2006).
[CrossRef]

L. I. Goray and S. Yu. Sadov, “Numerical modeling of coated gratings in sensitive cases,” in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), 365-379.

Saillard, M.

Sambles, J. R.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

Schattenburg, M. L.

Schmidt, G.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

J. Elschner, R. Hinder, F. Penzel, and G. Schmidt, “Existence, uniqueness and regularity for solutions of the conical diffraction problem,” Math. Models Meth. Appl. Sci. 10, 317-341 (2000).

G. Schmidt, “Boundary integral methods for periodic scattering problems,” in Around the Research of Vladimir Maz'ya II. Partial Differential Equations (Springer, 2010), pp. 337-364.

Seely, J. F.

Tsang, L.

L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetics Waves: Numerical Simulations (Wiley, 2001), pp. 61-110.

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R.Petit, ed. (Springer, 1980), pp. 101-121.
[CrossRef]

Wyrowski, F.

B. 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. Photonics News 10, August 1, 1999, pp. 44-48.
[CrossRef]

Zolla, F.

Adv. Comput. Math. (1)

J. Elschner, R. Hinder, and G. Schmidt, “Finite element solution of conical diffraction problems,” Adv. Comput. Math. 16, 139-156 (2002).
[CrossRef]

Appl. Opt. (3)

Comm. Comp. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Comm. Comp. Phys. 1, 984-1009 (2006).

IEEE Trans. Antennas Propag. (1)

J. P. Plumey, G. Granet, and J. Chandezon, “Differential covariant formalism for solving Maxwell's equations in curvilinear coordinates: oblique scattering from lossy periodic surfaces,” IEEE Trans. Antennas Propag. 43, 835-842 (1995).
[CrossRef]

J. Appl. Phys. (1)

L. I. Goray, J. F. Seely, and S. Yu. Sadov, “Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme ultraviolet gratings at near normal incidence,” J. Appl. Phys. 100, 094901 (2006).
[CrossRef]

J. Mod. Opt. (4)

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Figures (6)

Fig. 1
Fig. 1

Schematic conical diffraction by a grating.

Fig. 2
Fig. 2

Schematic diffraction by a simple grating in cross section.

Fig. 3
Fig. 3

Diffraction efficiencies of a highly conducting grating with c d = 0.5 and 2 H d = 0.3 , having the lamellar profiles slanted at an angle of 45° and hence being overhanging grooves, versus number of collocation points N. Other parameters are ϵ + = 1 , ϵ = ( 10 4 , 0 ) , μ ± = 1 , λ d = 0.8 , θ = 26.565 ° , ϕ = 14.478 ° , δ = 0 , and ψ = 0 .

Fig. 4
Fig. 4

Computation time for the example described in Table 2.

Fig. 5
Fig. 5

Diffraction efficiencies of a gold polygonal grating with 123 nodes, μ ± = 1 , and d = 200 nm for the incident wave with θ = 30 ° and ϕ = 88 ° : finite conductivity model ( δ = 34.143 ° and ψ = 0 ) and perfect conductivity model ( B z 0 : δ = 30.015 ° and ψ = 180 ° ) versus wavelength λ. Refractive indices of gold were taken from [38].

Fig. 6
Fig. 6

Average groove profile measured by atomic force microscopy.

Tables (6)

Tables Icon

Table 1 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Dielectric Lamellar Grating a

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Table 2 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Metallic Lamellar Grating a

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Table 3 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Dielectric Sine Grating for B z = 0 a

Tables Icon

Table 4 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Dielectric Sine Grating for E z = 0 a

Tables Icon

Table 5 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Metallic Echelette Grating for δ = 0 a

Tables Icon

Table 6 Diffraction Angles (θ, ϕ), Diffraction Efficiencies ( η ) , and Polarization Angles (δ, ψ) of a Metallic Echelette Grating for δ = 90 ° a

Equations (90)

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E i = p e i ( α x β y + γ z ) , H i = s e i ( α x β y + γ z ) ,
β > 0 , α 2 + β 2 + γ 2 = ω 2 ϵ + μ + ,
( α , β , γ ) = ω ϵ + μ + ( sin θ cos ϕ , cos θ cos ϕ , sin ϕ ) .
( E , H ) ( x , y , z ) = ( E , H ) ( x , y ) e i γ z ,
× E = i ω μ H , × H = i ω ϵ E ,
E T = E E z e z , H T = H H z e z ,
( ω 2 ϵ μ γ 2 ) E T = i γ E z + i ω μ × ( H z e z ) ,
( ω 2 ϵ μ γ 2 ) H T = i γ H z i ω ϵ × ( E z e z ) .
κ ( x , y ) = { ( ϵ + μ + ϵ + μ + sin 2 ϕ ) 1 2 = κ + ( x , y ) G + , ( ϵ μ ϵ + μ + sin 2 ϕ ) 1 2 = κ ( x , y ) G , }
( Δ + ω 2 κ 2 ) E z = ( Δ + ω 2 κ 2 ) H z = 0
[ ( n , 0 ) × E ] Σ × R = [ ( n , 0 ) × H ] Σ × R = 0 ,
[ E z ] Σ = [ H z ] Σ = 0 ,
[ γ ω 2 κ 2 t H z + ω ϵ ω 2 κ 2 n E z ] Σ = [ γ ω 2 κ 2 t E z ω μ ω 2 κ 2 n H z ] Σ = 0 .
[ E z ] Σ = [ H z ] Σ = 0 ,
[ ϵ n E z κ 2 ] Σ = ϵ + sin ϕ [ t B z κ 2 ] Σ ,
[ μ n B z κ 2 ] Σ = μ + sin ϕ [ t E z κ 2 ] Σ .
E z i ( x , y ) = p z e i ( α x β y ) , B z i ( x , y ) = q z e i ( α x β y )
with q z = ( μ + ϵ + ) 1 2 s z ,
u ( x + d , y ) = e i d α u ( x , y ) .
( E z , B z ) ( x , y ) = ( E z i , B z i ) + n Z ( E n + , B n + ) e i ( α n x + β n + y ) , y H ,
( E z , B z ) ( x , y ) = n Z ( E n , B n ) e i ( α n x β n y ) , y H ,
α n = α + 2 π n d , β n ± = ω 2 κ ± 2 α n 2
with 0 arg β n ± < π .
0 arg ϵ , arg μ π with arg ( ϵ μ ) < 2 π ,
E z = { u + + E z i u } , B z = { v + + B z i in G + , v in G , }
Δ u ± + ω 2 κ ± 2 u ± = Δ v ± + ω 2 κ ± 2 v ± = 0 in G ± ,
u = u + + E z i , ϵ n u κ 2 ϵ + n ( u + + E z i ) κ + 2 = ϵ + sin ϕ ( 1 κ + 2 1 κ 2 ) t v on Σ ,
v = v + + B z i , μ n v κ 2 μ + n ( v + + B z i ) κ + 2 = μ + sin ϕ ( 1 κ + 2 1 κ 2 ) t u on Σ ,
( u + , v + ) ( x , y ) = n = ( E n + , B n + ) e i ( α n x + β n + y ) for y H ,
( u , v ) ( x , y ) = n = ( E n , B n ) e i ( α n x β n y ) for y H .
σ ( t ) = ( X ( t ) , Y ( t ) ) ,
X ( t + 1 ) = X ( t ) + d ,
Y ( t + 1 ) = Y ( t ) , t R ;
Δ u + k 2 u = 0 with 0 arg k 2 < 2 π
Ψ k , α ( P ) = lim N i 2 d n = N N e i α n X + i β n | Y | β n , P = ( X , Y ) .
S φ ( P ) = 2 Γ φ ( Q ) Ψ k , α ( P Q ) d σ Q ,
D φ ( P ) = 2 Γ φ ( Q ) n ( Q ) Ψ k , α ( P Q ) d σ Q ,
u ( x , y ) = n = u n e i α n x + i β n | y | , | y | H ,
1 2 ( S n u D u ) = { u in G + 0 in G } ,
1 2 ( D S n u ) = { 0 in G + u in G }
( S φ ) + ( P ) = ( S φ ) ( P ) = V φ ( P ) ,
V φ ( P ) = 2 Γ Ψ k , α ( P Q ) φ ( Q ) d σ Q , P Σ .
( D φ ) + = ( K I ) φ , ( D φ ) = ( K + I ) φ
K φ ( P ) = 2 Γ φ ( Q ) n ( Q ) Ψ k , α ( P Q ) d σ Q + ( δ ( P ) 1 ) φ ( P ) .
( n S φ ) + = ( L + I ) φ , ( n S φ ) = ( L I ) φ ,
L φ ( P ) = 2 Γ φ ( Q ) n ( P ) Ψ k , α ( P Q ) d σ Q , P Σ .
t ( V φ ) ( P ) = 2 t Γ Ψ k , α ( P Q ) φ ( Q ) d σ Q , P Σ ,
t ( P ) ( P Q ) | P Q | 2 ,
J φ ( P ) = 2 lim δ 0 Γ \ Γ ( P , δ ) φ ( Q ) t ( P ) Ψ k , α ( P Q ) d σ Q = t ( V φ ) ( P ) ,
H φ ( P ) = 2 lim δ 0 Γ \ Γ ( P , δ ) φ ( Q ) t ( Q ) Ψ k , α ( P Q ) d σ Q ,
H φ ( P ) = 2 Γ Ψ k , α ( P Q ) t φ ( Q ) d σ Q = V ( t φ ) ( P ) , P Σ .
u + = 1 2 ( S + n u + D + u + ) ,
v + = 1 2 ( S + n v + D + v + ) in G + ,
E z i = 1 2 ( D + E z i S + n E z i ) ,
B z i = 1 2 ( D + B z i S + n B z i ) in G .
V + n ( u + + E z i ) ( I + K + ) ( u + + E z i ) = | 2 E z i | Σ ,
V + n ( v + + B z i ) ( I + K + ) ( v + + B z i ) = | 2 B z i | Σ ,
V ± φ ( P ) = 2 Γ φ ( Q ) Ψ ω κ ± , α ( P Q ) d σ Q , P Σ ,
u = S w , v = S τ
| u | Σ = V w , | n u | Σ = ( L I ) w ,
| v | Σ = V τ , | n v | Σ = ( L I ) τ ,
ϵ κ + 2 ϵ + κ 2 V + ( L I ) w ( I + K + ) V w sin ϕ ( 1 κ + 2 κ 2 ) V + t V τ = 2 E z i ,
μ κ + 2 μ + κ 2 V + ( L I ) τ ( I + K + ) V τ + sin ϕ ( 1 κ + 2 κ 2 ) V + t V w = 2 B z i .
V + t V = H + V = V + J
u + = 1 2 ( ϵ κ + 2 ϵ + κ 2 S + ( I L ) w + D + V w + sin ϕ ( κ 2 κ + 2 ) κ 2 S + J τ ) , u = S w ,
v + = 1 2 ( μ κ + 2 μ + κ 2 S + ( I L ) τ + D + V τ sin ϕ ( κ 2 κ + 2 ) κ 2 S + J w ) , v = S τ .
ϵ ϵ + κ 2 E z ¯ , μ μ + κ 2 B z ¯
Ω H ϵ ϵ + ( 1 κ 2 | E z | 2 ω 2 | E z | 2 ) + sin ϕ ( 1 κ + 2 1 κ 2 ) Γ t B z E z ¯ 1 κ + 2 Γ ( H ) n E z E z ¯ ϵ ϵ + κ 2 Γ ( H ) n E z E z ¯ = 0 ,
Ω H μ μ + ( 1 κ 2 | B z | 2 ω 2 | B z | 2 ) sin ϕ ( 1 κ + 2 1 κ 2 ) Γ t E z B z ¯ 1 κ + 2 Γ ( H ) n B z B z ¯ μ μ + κ 2 Γ ( H ) n B z B z ¯ = 0 ,
Γ ( H ) n E z E z ¯ = i β ( | E 0 + | 2 | p z | 2 + 2 i Im ( E 0 + p z ¯ e i β H ) ) + i n 0 β n + | E n + | 2 e 2 H Im β n + ,
Γ ( H ) n E z E z ¯ = i n Z β n | E n | 2 e 2 H Im β n ,
β κ + 2 | p z | 2 1 κ + 2 β n + > 0 β n + | E n + | 2 ϵ ϵ + κ 2 β n > 0 β n | E n | 2 = sin ϕ ( 1 κ + 2 1 κ 2 ) Im Γ t B z E z ¯ ,
β κ + 2 | q z | 2 1 κ + 2 β n + > 0 β n + | B n + | 2 μ μ + κ 2 β n > 0 β n | B n | 2 = sin ϕ ( 1 κ + 2 1 κ 2 ) Im Γ t E z B z ¯ ,
Im Γ t B z E z ¯ = Im Γ t E z B z ¯
| p z | 2 + | q z | 2 = β n + > 0 β n + β ( | E n + | 2 + | B n + | 2 ) + κ + 2 κ 2 β n > 0 β n β ( ϵ ϵ + | E n | 2 + μ μ + | B n | 2 ) .
R = β n + > 0 β n + β ( | E n + | 2 + | B n + | 2 )
T = β n > 0 β n β ( ϵ ϵ + | E n | 2 + μ μ + | B n | 2 ) .
Ω H G ϵ ϵ + ( 1 κ 2 | E z | 2 ω 2 | E z | 2 ) ϵ ϵ + κ 2 Γ ( H ) n E z E z ¯ = ϵ ϵ + κ 2 Γ n E z E z ¯ ,
Ω H G μ μ + ( 1 κ 2 | B z | 2 ω 2 | B z | 2 ) μ μ + κ 2 Γ ( H ) n B z B z ¯ = μ μ + κ 2 Γ n B z B z ¯ .
Im ϵ ϵ + κ 2 Γ n E z E z ¯ sin ϕ Im ( 1 κ + 2 1 κ 2 ) Γ t B z E z ¯ + β κ + 2 ( | E 0 + | 2 | p z | 2 ) + β n + > 0 β n + κ + 2 | E n + | 2 = 0 ,
Im μ μ + κ 2 Γ n B z B z ¯ + sin ϕ Im ( 1 κ + 2 1 κ 2 ) Γ t E z B z ¯ + β κ + 2 ( | B 0 + | 2 | q z | 2 ) + β n + > 0 β n + κ + 2 | B n + | 2 = 0 ,
| p z | 2 + | q z | 2 = β n + > 0 β n + β ( | E n + | 2 + | B n + | 2 ) + Im ϵ κ + 2 ϵ + κ 2 β Γ n E z E z ¯ + Im μ κ + 2 μ + κ 2 β Γ n B z B z ¯ sin ϕ β ( Im ( 1 κ + 2 κ 2 ) Γ ( t B z E z ¯ t E z B z ¯ ) ) = β n + > 0 β n + β ( | E n + | 2 + | B n + | 2 ) + κ + 2 β Im Γ ( ϵ ϵ + κ 2 n E z E z ¯ + μ μ + κ 2 n B z B z ¯ ) + 2 κ + 2 sin ϕ β Im 1 κ 2 Re Γ E z t B z ¯ .
| p z | 2 + | q z | 2 = R + A
A = κ + 2 β Im ( 1 κ 2 ( ϵ ϵ + Γ n E z E z ¯ + μ μ + Γ n B z B z ¯ + 2 sin ϕ Re Γ E z t B z ¯ ) ) .
A = κ + 2 β Im ( 1 κ 2 Γ ( ϵ ϵ + ( L I ) w V w ¯ + μ μ + ( L I ) τ V τ ¯ ) ) + 2 κ + 2 sin ϕ β Im 1 κ 2 Re Γ V w J τ ¯ .
w ( σ ( t ) ) e i α X ( t ) | σ ( t ) | w N ( t ) = k = N N a k e 2 π i k t ,
τ ( σ ( t ) ) e i α X ( t ) | σ ( t ) | τ N ( t ) = k = N N b k e 2 π i k t ,
V ± w ( σ ( t ) ) 2 e i α X ( t ) ( 0 1 log | 2 sin π ( t s ) | w N ( s ) d s + 0 1 g ± ( t , s ) w N ( s ) d s )
J ± w ( σ ( t ) ) e i α X ( t ) ( 0 1 cot π ( t s ) w N ( s ) d s + 0 1 j ± ( t , s ) w N ( s ) d s ) ,
p k ( t j ) = δ k j , k , j = 0 , , 2 N ,

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