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

Journal of the Optical Society of America A: Optics and Image Science, and Vision

Volumn 21, Issue 8, 2004, Pages 1529-1544

  Song, G Hugh ^a   Park, Mahn Yong ^a  

^a Gwangju Institute of Science and Technology (GIST)   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

BESSEL FUNCTIONS; COMPUTER SIMULATION; DIFFRACTION; FUNCTION EVALUATION; OPTICAL WAVEGUIDES; WAVEGUIDE COUPLERS;

ARRAYED WAVEGUIDES; DIRECTIONAL COUPLING; OPTIC STAR COUPLER; UNIFORM POWER SPLITTING;

OPTICAL BEAM SPLITTERS;

EID: 4043107741     PISSN: 10847529     EISSN: None     Source Type: Journal    
DOI: 10.1364/JOSAA.21.001529     Document Type: Article

References (29)

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20. As an interesting entry in the table of mathematical functions, χ∑s- ·χ exp(sα Js(κ sin(sβ + Θ)) = χ∑p=-χ exp(ipΘ Jp(κ sin(pβ - iα)) or, equivalently, χ∑s=1 sinh(sα)Js(κ sin(sβ + Θ)) = χ∑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.
21. See Ref. 15, Subsec. 11.3, Eqs. (2)-(6).
22. Although the angles of the triangle of Fig. 13 involved in the addition theorem are assumed to be real, the theorem was accepted to be valid for complex-valued angles. See Ref. 15, Subsec. 11.2, Eq. (1).
23. All the books on Bessel functions presume that k in Eq. (B1) are integers.
24. See Ref. 14, Subsec. 3.13.2.
25. It is a trivial generalization of the same formula for p = 1 appearing in Ref. 15, Subsec. 17.22, Eq. (3).
26. See Ref. 15, Subsec. 17.31, which cites F. W. Bessel, Berliner (1819), pp. 49-55.
27. See Ref. 15, Subsec. 17.22, Eq. (4): [-1]p-2[∂p-∂θp 1-1 - z cos ψ (z, θ)]0=0 = χ∑n=1 n2pJn(nz).
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.