Abstract

We propose a combination of ray optics and Fraunhofer multiple-slit diffraction theory for calculating the two-dimensional triangular periodic grating in the resonance domain. The peak of the envelope pattern of angular distribution of diffraction efficiency is calculated by ray optics while the peak width is calculated using Fraunhofer theory. It was clarified, using rigorous coupled-wave analysis and a nonstandard-finite-difference time-domain method, that the envelope pattern of the diffraction of the grating could be calculated easily and understood intuitively for the design of displays and lighting.

© 2009 Optical Society of America

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2008

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

2007

2006

2005

2004

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

2003

J. B. Cole and S. Banerjee, “Applications of nonstandard finite difference models to computational electromagnetics,” J. Differ. Equations 9, 1099-1112 (2003).
[CrossRef]

2001

2000

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

1999

1996

1994

1983

1982

1981

1978

1977

1962

Ahmed, S.

Banerjee, S.

T. Hoshino, S. Banerjee, M. Itoh, and T. Yatagai, “Design of a wavelength independent grating in the resonance domain,” Appl. Opt. 46, 7948-7962 (2007).
[CrossRef]

J. B. Cole and S. Banerjee, “Applications of nonstandard finite difference models to computational electromagnetics,” J. Differ. Equations 9, 1099-1112 (2003).
[CrossRef]

J. B. Cole and S. Banerjee, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” in Proceedings of The 23rd Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics, 2007), pp. 1531-1535.

J. B. Cole, S. Banerjee, and M. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E.Mickens, ed. (World Scientific, 2006), pp. 89-189.

Bogunovic, D.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

Cao, Y.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Chien, W.-T.

Choi, H. Y.

Cole, J. B.

J. B. Cole and S. Banerjee, “Applications of nonstandard finite difference models to computational electromagnetics,” J. Differ. Equations 9, 1099-1112 (2003).
[CrossRef]

J. B. Cole and S. Banerjee, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” in Proceedings of The 23rd Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics, 2007), pp. 1531-1535.

J. B. Cole, S. Banerjee, and M. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E.Mickens, ed. (World Scientific, 2006), pp. 89-189.

Feit, M. D.

Feldman, A.

Fleck, J. A.

Gao, H.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Gao, K.-F.

Gaylord, T. K.

Gilbert, L. R.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

Glytsis, E. N.

Haftel, M.

J. B. Cole, S. Banerjee, and M. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E.Mickens, ed. (World Scientific, 2006), pp. 89-189.

Harvey, J. E.

Horikx, J. J. L.

Hoshino, T.

T. Hoshino, S. Banerjee, M. Itoh, and T. Yatagai, “Design of a wavelength independent grating in the resonance domain,” Appl. Opt. 46, 7948-7962 (2007).
[CrossRef]

T. Hoshino, M. Itoh, and T. Yatagai, “An antireflective grating in the resonance domain for displays,” Appl. Opt. 46, 648-656 (2007).
[CrossRef] [PubMed]

Itoh, M.

T. Hoshino, M. Itoh, and T. Yatagai, “An antireflective grating in the resonance domain for displays,” Appl. Opt. 46, 648-656 (2007).
[CrossRef] [PubMed]

T. Hoshino, S. Banerjee, M. Itoh, and T. Yatagai, “Design of a wavelength independent grating in the resonance domain,” Appl. Opt. 46, 7948-7962 (2007).
[CrossRef]

Jiang, H. X.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

Jin, S. X.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

Kalil, K.

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

Krywonos, A.

Kwon, O. J.

Lagasse, P. E.

Lee, H.-S.

Lee, T.-X.

Li, J.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

Lin, C.-Y.

Lin, J. Y.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

Liu, D.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Ma, S.-H.

Marchiando, J. F.

Mellin, S.

Moharam, M. G.

Murty, M. V. R. K.

Nevitt, T. J.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

Nordin, G.

Nuijs, A. M.

Ouderkirk, A. J.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

Ouyang, M.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Park, S. R.

Roey, J. V.

Shi, J.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Shin, D.

Shingo, M.

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

Song, S.-H.

Spencer, G. H.

Stover, C. A.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

Sun, C.-C.

Sweatt, W. C.

Tatsuya, K.

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

Testorf, M.

Toshiyuki, M.

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

van der Donk, J.

Weber, M. F.

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

White, G. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

Yatagai, T.

T. Hoshino, S. Banerjee, M. Itoh, and T. Yatagai, “Design of a wavelength independent grating in the resonance domain,” Appl. Opt. 46, 7948-7962 (2007).
[CrossRef]

T. Hoshino, M. Itoh, and T. Yatagai, “An antireflective grating in the resonance domain for displays,” Appl. Opt. 46, 648-656 (2007).
[CrossRef] [PubMed]

Zhou, J.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. X. Jin, J. Li, J. Y. Lin, and H. X. Jiang, “InGaNOGaN quantum well interconnected microdisk light emitting diodes,” Appl. Phys. Lett. 77, 3236-3238 (2000).
[CrossRef]

IEICE Trans. Electron.

K. Kalil, M. Shingo, K. Tatsuya, and M. Toshiyuki, “Backlight unit with double surface light emission using a single micro-structured light-guide plate,” IEICE Trans. Electron. E87-C, 1954-1961 (2004).

J. Differ. Equations

J. B. Cole and S. Banerjee, “Applications of nonstandard finite difference models to computational electromagnetics,” J. Differ. Equations 9, 1099-1112 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Opt. Laser Technol.

M. Ouyang, Y. Cao, H. Gao, J. Shi, J. Zhou, and D. Liu, “Analysis on polarization dependence of Fraunhofer diffraction by metallic grating with short period,” Opt. Laser Technol. 40, 201-207 (2008).
[CrossRef]

Science

M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287, 2451-2456 (2000).
[CrossRef] [PubMed]

Other

J. B. Cole, S. Banerjee, and M. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes, R.E.Mickens, ed. (World Scientific, 2006), pp. 89-189.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

J. B. Cole and S. Banerjee, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” in Proceedings of The 23rd Annual Review of Progress in Applied Computational Electromagnetics (Applied Computational Electromagnetics, 2007), pp. 1531-1535.

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Figures (11)

Fig. 1
Fig. 1

Shape of the triangular grating profile and the definition of Λ, d, and d s . The fill factor is 0.5. The light direction for case A is shown; d s is assumed to be infinity. The refractive index n of the grating is 1.5.

Fig. 2
Fig. 2

Angular distribution of the diffraction efficiency for different polarization and light directions; Λ λ is 9.1 or 22.7. The aspect ratio is 1. Though the diffraction efficiency is discrete, it is connected with the auxiliary line to make it intelligible.

Fig. 3
Fig. 3

Angular distribution of the transmissive diffraction efficiency of the TE mode for case B for different Λ λ . The aspect ratio is 1. The diffraction efficiency is connected with the auxiliary line to make it intelligible; Λ λ is varied from 4.5 to 22.7.

Fig. 4
Fig. 4

Angular distribution of the transmissive diffraction efficiency of the TE mode for case A for different Λ λ . The aspect ratio is 2. The diffraction efficiency is connected with the auxiliary line to make it intelligible; Λ λ is varied from 4.5 to 22.7.

Fig. 5
Fig. 5

Angular distribution of the transmissive diffraction efficiency of the TE mode for different d Λ for case A. The diffraction efficiency is connected with the auxiliary line to make it intelligible. (a) d Λ is varied from 0.25 to 3. (b) d Λ is 2.5 and 3 and Λ λ is 9.1 and 22.7.

Fig. 6
Fig. 6

Incidence angle θ is varied. The angular distribution of θ from 0° to 30° and that of θ from 50° to 70° may be attributed to different groups by their peak width.

Fig. 7
Fig. 7

Angular distribution of the transmitted light is shown. The incidence angle is 20°, the direction of the incident light is case B, the polarization is TE, d Λ is 1, and Λ λ was varied. The angle of the peak is near 90°. The angular distribution varies greatly with Λ λ .

Fig. 8
Fig. 8

Field for FDTD calculation and the grating. The black rectangular area is expanded into the area of (b). (b) Phase distribution of the scattered light in the TE mode and for case A. There is only one groove, unlike Fig. 1. The width of the groove is 9.1 λ and the aspect ratio is 1.

Fig. 9
Fig. 9

Geometry of forward-diffracted wave vectors showing the conical nature of diffraction. Forward-diffracted waves ( i = 1 to i = + 2 ) are indicated by the arrow. The light travels from region 1 ( z < 0 ) to region 3 ( z > d ) .

Fig. 10
Fig. 10

Linearly polarized electromagnetic wave of wave vector k 1 .

Fig. 11
Fig. 11

Planar gratings resulting from the decomposition of the surface-relief grating into N thin gratings.

Tables (5)

Tables Icon

Table 1 Results of Curve Fitting for Fig. 3 by the Fraunhofer Single-Slit Diffraction Pattern for Different Λ λ in Case B a

Tables Icon

Table 2 Results of Curve Fitting for Fig. 4 by Fraunhofer Single-Slit Diffraction Pattern for Different Λ λ and d Λ in Case A a, b

Tables Icon

Table 3 Results of Curve Fitting for Fig. 5 by Fraunhofer Single-Slit Diffraction Pattern for Different Aspect Ratios in Case A a, b

Tables Icon

Table 4 Results of Curve Fitting for Fig. 6 by Fraunhofer Single-Slit Diffraction Pattern for Different Incidence Angles in Case A a

Tables Icon

Table 5 Total Transmissivity of Fig. 7 as a Function of Λ λ a

Equations (11)

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

I T ( θ ) = A 2 Λ 2 2 [ sin { π Λ 2 ( sin θ sin θ ) λ 2 } ( π Λ 2 ( sin θ sin θ ) λ 2 ) ] 2 ,
K = K x x ̂ + K z z ̂ = K sin ϕ x ̂ + K cos ϕ z ̂ .
E 3 = i T i exp [ j k 3 i ( r d z ̂ ) ] ,
λ d S x i ( z ) d z = j { 2 π i λ Λ sin ϕ S x i ( z ) ( ε I 1 2 sin α cos δ i λ Λ sin ϕ ) p a i p [ 2 π ε I 1 2 sin α sin δ U x p ( z ) ( 2 π ε I 1 2 sin α cos δ 2 π p λ Λ sin ϕ ) U y p ( z ) ] + 2 π U y i ( z ) } .
ε 1 ( x , z ) = h a h exp ( j h K r ) .
λ V ̇ = A V ,
S x i ( z ) = m C m ω 1 , i m exp ( λ m z λ ) ,
D E 1 i = Re [ ( k z 1 i ) ( k 1 cos α ) ] R i 2 ,
D E 3 i = Re [ ( k z 3 i ) ( k 1 cos α ) ] T i 2 ,
k z l i k = ε I 1 2 [ ( ε l ε I ) 2 ( sin α cos δ i λ Λ sin ϕ ) 2 ( sin α sin δ ) 2 ] 1 2 ,
n III sin ( θ i ) n I sin ( θ ) = i λ Λ .

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