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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 8 — Aug. 30, 2004
  • pp: 1529–1544

Bessel-function analysis of the optimized star coupler for uniform power splitting

G. Hugh Song and Mahn Yong Park  »View Author Affiliations

JOSA A, Vol. 21, Issue 8, pp. 1529-1544 (2004)

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An optimized N×N planar optic star coupler that utilizes directional coupling of arrayed waveguides for uniform power splitting is analyzed on the basis of special properties of the involved Bessel-function series. The analysis has provided a remarkably simple, novel basic design formula for such a device with much needed physical insights into the unique diffraction properties. For the analysis of diffraction from the end of directionally coupled arrayed waveguides, many useful formulas around the Bessel functions, such as the addition theorem and the Kepler–Bessel series, have been given in new forms.

© 2004 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(130.0130) Integrated optics : Integrated optics
(230.3120) Optical devices : Integrated optics devices
(230.7390) Optical devices : Waveguides, planar

Original Manuscript: December 15, 2003
Revised Manuscript: March 1, 2004
Manuscript Accepted: March 1, 2004
Published: August 1, 2004

G. Hugh Song and Mahn Yong Park, "Bessel-function analysis of the optimized star coupler for uniform power splitting," J. Opt. Soc. Am. A 21, 1529-1544 (2004)

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  20. As an interesting entry in the table of mathematical functions, ∑s=-∞∞exp(sα)Js(κ sin(sβ+Θ))=∑p=-∞∞exp(ipΘ)Jp(κ sin(pβ-iα))or, equivalently, ∑s=1∞sinh(sα)Js(κ sin(sβ+Θ))=∑p=1∞sin(pΘ)Jp(κ sin(pβ-iα)),with α, β, κ, and Θ real, would be more appropriate than Eq. (A7) = Eq. (A8). The criss-cross nature of the summation instigated by the rule of Eqs. (29) has brought in an identity formula that can hold only when summations over an infinite number of terms are carried out. From the literature surveyed, the above two formulas appear to have been newly found.
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  28. See Ref. 15, Eq. (1) in Subsec. 17.22, which cites Herz, Austrian Nach., CVII, 1884, columns 17–28.
  29. See Ref. 16, Chap. 9, formulas 9.1.27–28.

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