Abstract

Resonant subwavelength gratings (RSGs) offer narrowband high reflectivity with low-reflectivity sidebands. Analysis with the commonly used rigorous coupled-wave analysis assumes an RSG with infinite lateral extent and illumination by plane waves. This analysis is performed with a finite-difference semivectorial high-order accurate two-dimensional Helmholtz code that is able to simulate the entire finite RSG structure in the dimension of the grating vector. We study the effect of finite beam size on RSG reflectivity, resonant wavelength, and spectral response width. Independently, we study the effect of a finite RSG by varying the waveguide length and number of grating periods while fixing the beam size. We show that the placement of the waveguide end facets relative to the termination of the grating has a significant effect on the reflectivity and response width.

© 2004 Optical Society of America

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References

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    [CrossRef]

2001

2000

1998

G. R. Hadley, “Low-truncation-error finite difference representations of the 2-D Helmholtz equation,” Int. J. Electron. Commun. (AEU) 52, 310–316 (1998).

1997

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

S. M. Norton, T. Erdogan, G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1997).
[CrossRef]

1995

1993

1990

1989

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

1981

Avrutsky, I. A.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

Bagby, J. S.

Beck, W. A.

Bendickson, J. M.

Boye, R. R.

Brundrett, D. L.

Dunn, S. C.

Erdogan, T.

Friesem, A. A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Gao, X.

Gaylord, T. K.

Glytsis, E. N.

Hadley, G. R.

G. R. Hadley, “Low-truncation-error finite difference representations of the 2-D Helmholtz equation,” Int. J. Electron. Commun. (AEU) 52, 310–316 (1998).

D. W. Peters, S. A. Kemme, G. R. Hadley, “Low-sideband resonant subwavelength grating array design,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 290–295.

Jacob, D. K.

Kemme, S. A.

D. W. Peters, S. A. Kemme, G. R. Hadley, “Low-sideband resonant subwavelength grating array design,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 290–295.

Kostuk, R. K.

Magnusson, R.

Mait, J. N.

Mirotznik, M. S.

Moharam, M. G.

Morris, G. M.

Norton, S. M.

Peters, D. W.

D. W. Peters, S. A. Kemme, G. R. Hadley, “Low-sideband resonant subwavelength grating array design,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 290–295.

Prather, D. W.

Rosenblatt, D.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Sharon, A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Shi, S.

Sychugov, V. A.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

Wang, S. S.

Appl. Opt.

IEEE J. Quantum Electron.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

Int. J. Electron. Commun. (AEU)

G. R. Hadley, “Low-truncation-error finite difference representations of the 2-D Helmholtz equation,” Int. J. Electron. Commun. (AEU) 52, 310–316 (1998).

J. Mod. Opt.

I. A. Avrutsky, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

R. Magnusson, S. S. Wang, “Optical waveguide-grating filters,” in International Conference on Holography, Correlation Optics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–391 (1993).
[CrossRef]

D. W. Peters, S. A. Kemme, G. R. Hadley, “Low-sideband resonant subwavelength grating array design,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 290–295.

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Figures (15)

Fig. 1
Fig. 1

Representative RSG structure.

Fig. 2
Fig. 2

Reflectivity of an infinite RSG with normally incident, infinite-extent TE plane-wave excitation, calculated by use of RCWA.

Fig. 3
Fig. 3

Reflectivity of a plane wave from an infinite-extent RSG as a function of incident angle from the normal; dark color indicates high reflectivity.

Fig. 4
Fig. 4

Reflectivity of an infinite RSG as a function of Gaussian beam width (width measured between 1/e2 points of the beam intensity) determined with a weighted-RCWA technique and with the finite-difference Helmholtz code. FDM, finite-difference method.

Fig. 5
Fig. 5

Configuration for FDM Helmholtz computations.

Fig. 6
Fig. 6

Spectral response of RSG for various Gaussian beam widths calculated with the FDM Helmholtz equation.

Fig. 7
Fig. 7

Comparison of the spectral response of a 200-μm Gaussian beam width by use of RCWA with plane-wave decomposition and the FDM Helmholtz method with a 1.22-mm-wide grating.

Fig. 8
Fig. 8

Simulation format used to determine the reflectivity of finite grating RSG.

Fig. 9
Fig. 9

Reflectivity from several finite grating widths with a fixed RSG size of 1.22 mm and a fixed incident 800-μm-width Gaussian.

Fig. 10
Fig. 10

Spectral peak reflectivity as a function of grating width for the case of fixed beam size. Note that as grating width decreases the beam overfills the grating, and at the lower limit the beam reflects off thin-film layers with no grating present.

Fig. 11
Fig. 11

Spectral FWHM of reflectivity response as a function of grating width for the case of fixed beam size and fixed RSG size. (Note: The fitted curve is an exponential with α=0.039 μm-1.)

Fig. 12
Fig. 12

Finite grating and finite waveguide layer (boundary conditions identical to Fig. 5).

Fig. 13
Fig. 13

Peak reflectivity as a function of the extension of the finite waveguide past the termination of the finite grating for five grating widths.

Fig. 14
Fig. 14

Reflectivity from an RSG with the waveguide terminated at the end of the grating compared with the reflectivity from the same RSG with the waveguide extended to infinity.

Fig. 15
Fig. 15

FWHM as a function of the grating width for a finite grating on an infinite waveguide and a finite grating on a waveguide terminated at the last grating ridge.

Equations (3)

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E(x)=A exp[-(x/a)]2,
F(θ, λ)=Aπa exp(πaθ/λ)2.
R=F2(θ, λ)R(θ)dθ.

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