Abstract

Broadening of the angular response of two-dimensional (2D) guided mode resonant spectral filters at oblique incidence is investigated. Coupling into multiple fundamental guided resonant modes having the same propagation constant but propagating in different planes (inherent multiple-plane diffraction by 2D gratings) is shown to significantly broaden the angular tolerance while maintaining narrow linewidth. Resonances have symmetric and broad angular responses when the incident wave is coupled to four resonant modes in a structure with a hexagonal grating pattern. Further broadening is implemented by enhancing the second Bragg diffraction of the 2D grating structure. Resonance with a narrow spectral linewidth (ΔλFWHM1.6×104λ0) and angularly tolerant to an 6μm beam diameter is obtained. A second approach utilizing a dual 2D grating configuration with a second grating on the substrate side is shown to increase the lateral confinement, causing the merging of two successive resonant bands. This results in further improvement of the angular∕spectral linewidth ratio by 80%.

© 2007 Optical Society of America

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References

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2006 (1)

2005 (1)

A.-L. Fehrembach and A. Sentenac, "Unpolarized narrow-band filtering with resonant gratings," Appl. Phys. Lett. 86, 121105 (2005).
[CrossRef]

2004 (2)

2003 (1)

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonics devices enabled by waveguide-mode resonance effects in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef] [PubMed]

2001 (2)

2000 (2)

D. K. Jacob, S. C. Dunn, and M. G. Moharam, "Design considerations for narrow-band dielectric resonant grating reflection filters of finite length," J. Opt. Soc. Am. A 17, 1241-1249 (2000).
[CrossRef]

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proc. SPIE 3911, 86-94 (2000).
[CrossRef]

1998 (1)

F. Lemarchand, A. Sentenac, and H. Giovannini, "Increasing the angular tolerance of resonant grating filters with doubly periodic structures," Opt. Lett. 22, 1149-1151 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (2)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 5, 1077-1086 (1995).
[CrossRef]

1992 (1)

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

1965 (1)

Appl. Opt. (3)

Appl. Phys. Lett. (2)

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

A.-L. Fehrembach and A. Sentenac, "Unpolarized narrow-band filtering with resonant gratings," Appl. Phys. Lett. 86, 121105 (2005).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Lett. (2)

F. Lemarchand, A. Sentenac, and H. Giovannini, "Increasing the angular tolerance of resonant grating filters with doubly periodic structures," Opt. Lett. 22, 1149-1151 (1998).
[CrossRef]

D. K. Jacob, S. C. Dunn, and M. G. Moharam, "Interference approach applied to dual-grating dielectric resonant grating reflection filters," Opt. Lett. 26, 1749-1751 (2001).
[CrossRef]

Proc. SPIE (2)

D. Wawro, S. Tibuleac, R. Magnusson, and H. Liu, "Optical fiber endface biosensor based on resonances in dielectric waveguide gratings," Proc. SPIE 3911, 86-94 (2000).
[CrossRef]

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonics devices enabled by waveguide-mode resonance effects in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef] [PubMed]

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Figures (15)

Fig. 1
Fig. 1

(a) Spectral and (b) angular responses of resonances due to the single-propagation and counterpropagation resonant modes.

Fig. 2
Fig. 2

Configurations of diffraction∕guidance planes at oblique incidence and phase-matching diagram of resonant modes with the same β in the 2D GMR with (a) rectangular-, (b) hexagonal-lattice grating.

Fig. 3
Fig. 3

(Color online) Configurations of diffraction∕guidance plane and phase-matching diagram of three resonant modes at off-resonant angle (a) on-, (b) off-incident plane.

Fig. 4
Fig. 4

(Color online) Configurations of diffraction∕guidance plane and phase-matching diagram of four resonant modes at off-resonant angle (a) on-, (b) off-incident plane.

Fig. 5
Fig. 5

Two-dimensional GMR structure configuration.

Fig. 6
Fig. 6

(a) Phase-matching conditions (b) spectral (c) and (d) angular resonant response along and perpendicular to plane of incidence of 2D GMRs in Table 1.

Fig. 7
Fig. 7

(Color online) (a) Scattering of four resonant modes at resonance (A, forward diffraction; B, a crossed diffraction) and dephasing manners of a crossed diffraction at off-resonance (b) θ 0 + Δ θ , (c) φ 0 + Δ φ .

Fig. 8
Fig. 8

Grating structure η B and η d of GMR 2 in Table 1 versus the grating depth, where the grating has r = 0.13 μ m , λ 0 = 0.98 μ m , θ inc 10.4 ° .

Fig. 9
Fig. 9

(a)–(e) Angular responses of four TM 0 resonant modes located at λ 0 = 0.98 μ m , θ inc = 10.4 ° , and dispersion plots calculated by RCWA (a)–(b) t g = 0.05 μ m , t f = 0.503 μ m , and (d)–(e) when t g = 0.15 , t f = 0.44 μ m , (c) dispersion plot calculated by HW approach for t g = 0.05 μ m .

Fig. 10
Fig. 10

Resonant band diagram of the structure in Table 2 calculated by RCWA and angular responses at λ 0 , (a)–(d) t g = 0.25 , (b)–(e) t g = 0.35 , and (c)–(f) t g = 0.55 μ m .

Fig. 11
Fig. 11

(Color online) Angular-wavelength linewidth ratio versus grating depth (Table 2).

Fig. 12
Fig. 12

Dual-grating 2D GMR configuration.

Fig. 13
Fig. 13

(Color online) (a) Scattering of four resonant modes at resonance (A, forward diffraction; B, a crossed diffraction by the substrate grating) and dephasing manners of a crossed diffraction (B) at off-resonance (b) θ 0 + Δ θ , (c) φ 0 + Δ φ .

Fig. 14
Fig. 14

(Color online) Angular-wavelength linewidth ratio versus substrate grating depth, t g2 (Table 3).

Fig. 15
Fig. 15

Resonant band diagram of the structure in Table 4 calculated by RCWA (a) t g 2 = 0.15 μ m , (b) t g 2 = 0.25 μ m , (c) t g 2 = 0.34 μ m , (d) plot of angular responses at λ 0 of the structures with t g 2 = 0.25 0.4 μ m .

Tables (3)

Tables Icon

Table 1 Spectral and Angular Linewidth of Resonances ( λ 0 = 0.98 μm) by a Counterpropagation Resonant Mode in 2D GMR at Oblique Incidence

Tables Icon

Table 2 Spectral and Angular Linewidth of Resonances ( λ 0 = 0.98 μm) in a Hexagonal-Lattice Grating Resonance Structures (Λ a = 0.55, Λ b = 0.6, and r = 0.13 μm) at Oblique Incidence ( φ 0 = 90°, ψ 0 = 0°)

Tables Icon

Table 3 Spectral and Angular Linewidth of Resonances ( λ 0 = 0.98 μm) in a Dual Hexagonal-Lattice Grating Resonance Structures (Λ a = 0.55, Λ b = 0.6 μm, Superstrate Grating: nH∕nL = 2∕1, t g 1 = 0.55 μm, r 1 = 0.13 μm, Substrate Grating: nH∕nL = 2∕1.47, r 2 = 0.21 μm) at Oblique Incidence ( φ 0 = 90°, ψ 0 = 0°, φ 0 = 14°)

Equations (11)

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{ β } k diff , 1 st ,
Δ θ FWHM Δ λ FWHM Λ   cos   θ 0
Δ ϕ + 1 Δ θ Δ ϕ 1 Δ θ 2 { κ Δ β β Δ κ κ 2 ( t f + 1 γ c + 1 γ c ) + β κ ( Δ γ c γ c 2 Δ γ s γ s 2 ) } k 0   cos   θ inc .
k x , p q = k x ,inc ( p 2 π Λ b   sec ( ζ ) + q 2 π Λ a   tan ( ζ ) ) ,
k y , p q = k y ,inc ( q 2 π Λ a ) ,
sin   θ inc = λ 0 2 Λ b [ 1 ( Λ b / Λ a ) 2 ] .
sin   θ inc = λ 0 2 Λ a [ 1 3 4 ( Λ b / Λ a ) 2 1 ] .
| d ϕ + d θ inc | k 0   sin   θ inc β k 0   cos   θ inc | d ϕ d β | ,
| d ϕ d θ inc | k 0   cos   θ inc | d ϕ d β | .
| d ϕ + d θ inc | = k 0   sin   θ inc β k 0   cos   θ inc | d ϕ d β | ,
| d ϕ d θ inc | = | k 0   sin   θ inc ( 2 π / Λ a ) | β k 0   cos   θ inc | d ϕ d β | .

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