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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 2, Iss. 6 — Jun. 1, 1985
  • pp: 863–871

Analysis of waveguide gratings: application of Rouard’s method

L. A. Weller-Brophy and D. G. Hall  »View Author Affiliations

JOSA A, Vol. 2, Issue 6, pp. 863-871 (1985)

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This paper describes the adaptation of Rouard’s method, a familiar computational technique used in thin-film coating design, to the analysis of waveguide diffraction gratings. The approach can be used to generate the spectral and the angular characteristics of arbitrary straight-line gratings (periodic or nonperiodic), is easy to implement on a computer, and provides an intuitively appealing picture of the operation of such gratings. The application of the thin-film method is extended to periodic gratings as well as to gratings having either a linear or a quadratic chirp in period. Specific examples are used to compare the results of the thin-film computational method with those predicted by previous authors.

© 1985 Optical Society of America

Original Manuscript: September 26, 1984
Manuscript Accepted: February 14, 1985
Published: June 1, 1985

L. A. Weller-Brophy and D. G. Hall, "Analysis of waveguide gratings: application of Rouard’s method," J. Opt. Soc. Am. A 2, 863-871 (1985)

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  28. Typically, the coupled-mode analysis of a Bragg-matched periodic grating of length L yields the grating amplitude reflectivity, r = i tanh(KL). In such treatments, the surface corrugation of the grating is described byx=h+Δhcos(2πΛz),0≤z≤L.In this paper, we have specified the grating corrugation in the slightly different form [see Eq. (4)]:x=h+Δhsin(2πΛz),0≤z≤L.When this form is used in the coupled-mode equations, the resulting grating amplitude reflectivity is given byr=−tanh(KL).For the case of a detuned periodic grating, the amplitude reflectivity is written in the more general formr=−Ksinh(αL)αcosh(αL)−iδsinh(αL).
  29. We consider only TE–TE coupling in the examples presented in this paper and use the TE–TE coupling coefficient derived by Wagatsuma et al.using a coupled-mode analysis.14,15 It should be noted that there is little agreement among the coupling coefficients presented in the literature, 2–5,8,12–15 a point that has been discussed by several authors.1,4,12 Consequently, a judicious choice of coupling coefficient is required in order to obtain the most accurate predictions of the grating-response characteristics.
  30. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Department of Commerce, National Bureau of Standards, Washington, D.C., 1972), Eq. (4.5.26).
  31. When propagating in the +z direction, the grating corrugation is described byx=h+Δhsin(2πΛz),0≤z≤Λ,resulting in the amplitude reflectivity r = −tanh(KΛ). When propagating in the −z direction, the grating appears reversed and is described byx=h−Δsin(2πΛz),0≤z≤Λ.The change in sign of the sinusoidal corrugation results in a corresponding change in the grating-period reflectivity to r = +tanh(KΛ).Since a third-order reflection consists of two reflections of the forward-propagating mode and one reflection of the backward-propagating mode, the sign of a third-order reflection is positive.
  32. The method of Hong et al.17 must be modified to predict the width of the grating response as well as the amplitude. We have done this by ascribing a reflectivity of zero percent for spectral or angular components having propagation constants β not satisfyingβL≤β≤β0,whereβL=πcos(θ)Λ(z=L)andβ0=πcos(θ)Λ(z=0).
  33. The effective-length approximation16,18 takes a slightly different form in the two references cited. The formulation presented by Fukuzawa and Nakamura16 is used throughout this paper.

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