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

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

http://dx.doi.org/10.1364/JOSAA.21.001529

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### Abstract

An optimized

© 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

**History**

Original Manuscript: December 15, 2003

Revised Manuscript: March 1, 2004

Manuscript Accepted: March 1, 2004

Published: August 1, 2004

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-8-1529

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### References

- C. Dragone, “Efficient N×N star couplers using Fourier optics,” J. Lightwave Technol. 7, 479–489 (1989). [CrossRef]
- C. Dragone, C. H. Henry, I. P. Kaminow, R. C. Kistler, “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photonics Technol. Lett. 1, 241–243 (1989). [CrossRef]
- C. Dragone, “Optimum design of a planar array of tapered waveguides,” J. Opt. Soc. Am. A 7, 2081–2093 (1990). [CrossRef]
- K. Okamoto, H. Takahashi, S. Suzuki, A. Sugita, Y. Ohmori, “Design and fabrication of integrated-optic 8×8 star coupler,” Electron. Lett. 27, 774–775 (1991). [CrossRef]
- M. Y. Park, G. H. Song, K. Hwang, H. J. Lee, K.-B. Chung, “Design of waveguide-grating routers with minimal insertion-loss over all channels based on coupling between adjacent waveguides,” in Conference on Lasers and Electro-Optics, Vol. 56 of 2001 OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 127–128.
- Y. P. Li, C. H. Henry, “Silicon optical bench waveguide technology,” in Optical Fiber Telecommunications, I. P. Kaminow, T. L. Koch, eds. (Academic, San Diego, Calif., 1997), Vol. IIIB, Chap. 8.
- C. R. Doerr, “Planar lightwave devices for WDM,” in Optical Fiber Telecommunications, I. P. Kaminow, T. Li, eds. (Academic, San Diego, Calif., 2002), Vol. IVA, Chap. 9.
- S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973). [CrossRef]
- G. H. Song, W. J. Tomlinson, “Fourier analysis and synthesis of adiabatic tapers in integrated optics,” J. Opt. Soc. Am. A 9, 1289–1300 (1992). [CrossRef]
- G. H. Song, “Principles of photonics I, theory of lightwave propagation,” Class note available from the author. A plot for this assertion is given in the class notes based on the theory of D. Marcuse, Ref. 11, Sec. 6.2.
- D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).
- C. Dragone, “Planar waveguide array with nearly ideal radiation characteristics,” Electron. Lett. 38, 880–881 (2002). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
- W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enl. ed. (Springer, New York, 1966), Subsec. 3.2.1.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944).
- M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
- See Ref. 16, F. W. J. Olver, “Bessel functions of integer order,” Chap. 9, formula 9.3.1 and Fig. 9.3.
- K. Okamoto, Fundamentals of Optical Waveguides (Academic, San Diego, Calif., 2000).
- J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997). [CrossRef]
- 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.
- See Ref. 15, Subsec. 11.3, Eqs. (2)–(6).
- Although the angles of the triangle of Fig. 13involved 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).
- All the books on Bessel functions presume that kin Eq. (B1) are integers.
- See Ref. 14, Subsec. 3.13.2.
- It is a trivial generalization of the same formula for p=1appearing in Ref. 15, Subsec. 17.22, Eq. (3).
- See Ref. 15, Subsec. 17.31, which cites F. W. Bessel, Berliner (1819), pp. 49–55.
- See Ref. 15, Subsec. 17.22, Eq. (4): [-1]p2∂p∂θp11-z cos ψ(z, θ)0=0=∑n=1∞n2pJn(nz).
- See Ref. 15, Eq. (1) in Subsec. 17.22, which cites Herz, Austrian Nach., CVII, 1884, columns 17–28.
- See Ref. 16, Chap. 9, formulas 9.1.27–28.

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