Abstract

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

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References

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  1. C. Dragone, “Efficient N×N star couplers using Fourier optics,” J. Lightwave Technol. 7, 479–489 (1989).
    [CrossRef]
  2. 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]
  3. C. Dragone, “Optimum design of a planar array of tapered waveguides,” J. Opt. Soc. Am. A 7, 2081–2093 (1990).
    [CrossRef]
  4. 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]
  5. 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.
  6. 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.
  7. 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.
  8. 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]
  9. 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]
  10. 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.
  11. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).
  12. C. Dragone, “Planar waveguide array with nearly ideal radiation characteristics,” Electron. Lett. 38, 880–881 (2002).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  14. 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.
  15. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944).
  16. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  17. See Ref. 16, F. W. J. Olver, “Bessel functions of integer order,” Chap. 9, formula 9.3.1 and Fig. 9.3.
  18. K. Okamoto, Fundamentals of Optical Waveguides (Academic, San Diego, Calif., 2000).
  19. J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997).
    [CrossRef]
  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.
  21. See Ref. 15, Subsec. 11.3, Eqs. (2)–(6).
  22. 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).
  23. All the books on Bessel functions presume that kin Eq. (B1) are integers.
  24. See Ref. 14, Subsec. 3.13.2.
  25. It is a trivial generalization of the same formula for p=1appearing 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]p2∂p∂θp11-z cos ψ(z, θ)0=0=∑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.

2002 (1)

C. Dragone, “Planar waveguide array with nearly ideal radiation characteristics,” Electron. Lett. 38, 880–881 (2002).
[CrossRef]

1997 (1)

J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997).
[CrossRef]

1992 (1)

1991 (1)

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]

1990 (1)

1989 (2)

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]

1973 (1)

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]

1884 (1)

See Ref. 15, Eq. (1) in Subsec. 17.22, which cites Herz, Austrian Nach., CVII, 1884, columns 17–28.

1819 (1)

See Ref. 15, Subsec. 17.31, which cites F. W. Bessel, Berliner (1819), pp. 49–55.

Bessel, F. W.

See Ref. 15, Subsec. 17.31, which cites F. W. Bessel, Berliner (1819), pp. 49–55.

Chen, J. C.

J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997).
[CrossRef]

Chung, K.-B.

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.

Doerr, C. R.

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.

Dragone, C.

C. Dragone, “Planar waveguide array with nearly ideal radiation characteristics,” Electron. Lett. 38, 880–881 (2002).
[CrossRef]

J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997).
[CrossRef]

C. Dragone, “Optimum design of a planar array of tapered waveguides,” J. Opt. Soc. Am. A 7, 2081–2093 (1990).
[CrossRef]

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]

Garmire, E.

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]

Garvin, H. L.

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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Henry, C. H.

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]

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.

Herz,

See Ref. 15, Eq. (1) in Subsec. 17.22, which cites Herz, Austrian Nach., CVII, 1884, columns 17–28.

Hunsperger, R. G.

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]

Hwang, K.

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.

Kaminow, I. P.

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]

Kistler, R. C.

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]

Lee, H. J.

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.

Li, Y. P.

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.

Magnus, W.

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.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, New York, 1991).

Oberhettinger, F.

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.

Ohmori, Y.

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]

Okamoto, K.

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]

K. Okamoto, Fundamentals of Optical Waveguides (Academic, San Diego, Calif., 2000).

Olver, F. W. J.

See Ref. 16, F. W. J. Olver, “Bessel functions of integer order,” Chap. 9, formula 9.3.1 and Fig. 9.3.

Park, M. Y.

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.

Somekh, S.

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]

Song, G. H.

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.

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.

Soni, R. P.

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.

Sugita, A.

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]

Suzuki, S.

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]

Takahashi, H.

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]

Tomlinson, W. J.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944).

Yariv, A.

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]

Appl. Phys. Lett. (1)

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]

Austrian Nach. (1)

See Ref. 15, Eq. (1) in Subsec. 17.22, which cites Herz, Austrian Nach., CVII, 1884, columns 17–28.

Berliner (1)

See Ref. 15, Subsec. 17.31, which cites F. W. Bessel, Berliner (1819), pp. 49–55.

Electron. Lett. (3)

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]

C. Dragone, “Planar waveguide array with nearly ideal radiation characteristics,” Electron. Lett. 38, 880–881 (2002).
[CrossRef]

J. C. Chen, C. Dragone, “Waveguide grating routers with greater channel uniformity,” Electron. Lett. 33, 1951–1952 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

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]

J. Lightwave Technol. (1)

C. Dragone, “Efficient N×N star couplers using Fourier optics,” J. Lightwave Technol. 7, 479–489 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

Other (19)

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).

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.

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).

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).

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. 16, Chap. 9, formulas 9.1.27–28.

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of an N×N star coupler. FPR, the free-propagation region; WOR, waveguide-opening region. Not drawn to scale.

Fig. 2
Fig. 2

Plot for field propagation through a star coupler of the optimized design that utilizes directional coupling in the input array by the wide-angle finite-difference-based beam propagation method. The FPR starts at z=0.

Fig. 3
Fig. 3

Three diagrams for explaining miscellaneous phase corrections to the propagating field through the star coupler utilizing directional coupling in the input waveguide array.

Fig. 4
Fig. 4

Three variations of coupled arrayed waveguides: (a) and (b) variations with the gap distances of 4 and 1 μm, respectively; (c) variation with the converging array of waveguides with the gap distance becoming zero in the end. The convergence angle is derived from the actual configuration of the optimized design of the star coupler.

Fig. 5
Fig. 5

One-sided plots for the result of beam propagation simulation in solid curves for waveguide arrays of the three configurations in Fig. 4, compared with the dotted curves obtained by formula (7) in the text with x=3.1 and ϕ^1(y) of the standard Gaussian function: (a) parallel array with gap distance 4 μm, (b) parallel array with gap distance 1 μm, and (c) converging array of Fig. 4(c) with the circled opening of radius z=1.8 mm. All three configurations effectively yield x=3.1.

Fig. 6
Fig. 6

Plot of |ϕ| in expression (18) with respect to Θk¯hy/R(z) for Fresnel diffraction patterns from the source of a series of deltalike patterns weighted by {Jm(2x)}, x=3.1.

Fig. 7
Fig. 7

Fresnel diffraction patterns at several distances by use of renormalized scale Θ with a set of near-optimized design parameters in Eq. (10).

Fig. 8
Fig. 8

Various intensity plots for g(Θ, Z)|ϕ˘(y, z)|2 in expression (41) with several expressions in expression (A1) for |ϕ˘(y, z)|2. Solid curve, the paraxial approximation itself from the absolute square of Eq. (20); this curve virtually coincides with the one for gs=-fs(+2) and almost with gp=-fp(-T), and gp=-fp(-1)=gs=-fs(+1) virtually coincides with gp=-fp(-2).

Fig. 9
Fig. 9

Plot for p=-[(2gfp(-1))/Θ2]Θ=0p=-[(2gfp(-2))/Θ2]Θ=0 (dashed curve) and p=-[(2gfp(-T))/Θ2]Θ=0 (solid curve) with 1/ζZ/xΘw2) as the horizontal axis. Θw=3.0 has been chosen for the plot. Note that the zero crossing happens at 1/ζ1.25. From the convergence range of the Kapteyn series and the Kepler–Bessel series, the curves are valid for 1/ζ>1.

Fig. 10
Fig. 10

Flattened diffraction patterns |gp=-fp(-T)|2, at Z=Z, obtained by several choices for the design parameters Θw=3.0, π, 3.4, 4.0, 1.5π, 2π. The normalization factor |q(0)/q(z)| has been deliberately omitted to effectively show the flatness of the patterns near the top, which is responsible for the increased plateau for higher Θw.

Fig. 11
Fig. 11

Plot of two curves for three sets of data. The dashed curve is obtained by expression (41) with expression (46), as well by the standard paraxial formula, with the input field taken from the dashed curve in Fig. 5(c) expressed by expression (7) with Eqs. (9). The virtual coincidence confirms the validity of our analytic result. The solid curve is the one obtained by the paraxial formula at z=1800 μmz with the light input field taken from the solid curve in Fig. 5(c), obtained from the light input launched to the center of the waveguide array with Θw=3.0π in Fig. 4(c).

Fig. 12
Fig. 12

Comparison of the approximate formulas. Dashed curve, gp=-fp(-2) approximated by expression (A9) for the original Eq. (A2) of a solid curve. Dashed–dotted curve, gs=-fs(+2) approximated by expression (A10) for the original Eq. (A3) of a dotted curve.

Fig. 13
Fig. 13

Triangle depicted for a special case of the addition theorem, i.e., u=v.

Tables (1)

Tables Icon

Table 1 Parameters for One Exemplary Optimized Design

Equations (127)

Equations on this page are rendered with MathJax. Learn more.

ϕ(y, z)m=-M/2M/2am(z)ϕ^1(y-mh),
-|ϕ^1(y)|2dy=1.
ddz am(z)=iβam(z)+iKam-1(z)+iKam+1(z),
ddz Vm(z)=iKVm-1(z)+iKVm+1(z).
Vm(-Lc)=δm0,
Vm(0)=imJm(2x),
ϕ(y, 0)m=-imJm(2x)ϕ^1(y-mh).
P-|ϕ(y, 0)|2dym=-im-mJm2(2x)-|ϕ^1(y-mh)|2dy=1,
ϕ^1(y)=A exp(-y2/w02),A(2/πw02)1/4
w0=4.0μm
x3.1
x-0K(z)dz.
exp(im2h2β/2z)
ϕ(y, z)k¯2π|ρ|1/2z|ρ|expik¯|ρ|-iπ4×Aπw0exp(-k¯2w02y2/4z2)×m=-imJm(2x)expim k¯hyz
ϕ(y, z)k¯2|ρ|1/2z|ρ|expik¯|ρ|-iπ4Aw0×exp(-[k¯/2z]2w02y2+i2x cos(k¯hy/z)),
ϕ(y, z)k¯/2πz)expik¯z-iπ4m=-imJm(2x)×-ϕ^1(y-mh)expik¯[y-y]22zdy.
ϕ^1(y)=δ(y)
ϕ(y, z)k¯/2πzexp(ik¯z-iπ/4)×m=-imJm(2x)expik¯[y-mh]22z=k¯/2πzexp(ik¯z-iπ/4+ik¯y2/z)×m=-imJm(2x)expim2k¯h22z-imh k¯z y.
θ12π/k¯h
ϕ(y, z)=Aq(0)/q(z)exp(ik¯z)×m=-imJm(2x)expik¯[y-mh]22q(z),
1/q(z)1/R(z)+i2/k¯w2(z),
R(z)[1+z02/z2]z,
w2(z)[1+z2/z02]w02,
z0w02k¯/2.
ϕ(y, z)?Aq(0)/q(z)exp(ik¯z-y2/w2(z))
×m=-imJm(2x)expik¯[y-mh]22R(z)
θww0/z02/k¯w0.
|ϕ(y, z)|2q(0)q(z)A2exp-2y2w2(z)|ϕ˘(y, z)|2,
|ϕ˘(y, z)|2m=-l=-im-lJm(2x)Jl(2x)×exp-2h2w2(z)m2+l22-yh [m+l]+ik¯h22R(z)[m2-l2]-2yh [m-l]
m-lp,m+ls.
m=p+s/2,l=s/2-p.
m2+l2=[m+l]2+[m-l]22,
φ˘(y,z)2=s=-exp(-s2h22w2(z)+2shyw2(z)) ×p=-i2pJs/2+p(2x)Js/2-p(2x) ×exp(-p22h2 w2(z)-ip[2y-sh]k¯hR(z) ),
2[ph]2/w2(z)[xh/w(z)]21,
exp(-p22h2/w2(z))1,
φ˘(y,z)2s=-isexp(-s2h22w2(z)+2shyw2(z)) ×p=-Jp+s/2(2x)Jp-s/2(2x) ×exp(ip[sh-2y]k¯h/R(z))s=-isexp(-s2h22w2(z)+2shyw2(z)) ×i-sJ-s(4xsin(sk¯h2 2R(z)-k¯hR(z)y) ) =s=-exp(-s2h22w2(z)-2shyw2(z)) ×Js(-4xsin(sk¯h2 2R(z)+k¯hR(z)y) ),
s[m+l]/2,
φ˘(y,z)2=p=-ipexp(-p2h22w2(z)-ipk¯hR(z)y) ×s=-Js+p/2(2x)Js-p/2(2x) ×exp(-s22h2 w2(z)+ipsk¯h2 R(z)+4shyw2(z) ) p=-exp(-p2h22w2(z)+ipk¯hR(z)y) ×Jp(-4xsin(pk¯h2 2R(z)+i2hyw2(z) ) ),
Z(z)z/z0.
Θ(y, z)k¯hy/R(z).
Θw2(Z)4h2/w021+Z-2.
exp-2y2w2(z)|ϕ˘(y, z)|2 g(Θ, Z)p=-fp(-T)(Θ, Z),
g(Θ, Z)exp(-2Θ2/Θw2(Z)).
z0=47μm,Z=38,Θw2(Z)Θw2()=9.0.
exp(-p2Θw2/8Z2)1-p2Θw2/8Z2,
exp(-s2Θw2/2Z2)1-s2Θw2/2Z2.
fp(-T)(Θ, Z)1-p2Θw28Z2exp(ipΘ)×Jp-[pΘw2+i4Θ] xZ-Θw22Z2 xΘw24Z p+iΘZ×Jp-[pΘw2+i4Θ] xZ+Θw22Z2 4x2Jp-[pΘw2+i4Θ] xZexp(ipΘ)1-p2Θw28Z2×Jp-[pΘw2+i4Θ] xZ+2Θw2xZ2Jp-[pΘw2+i4Θ] xZ,
2g|ϕ˘|2Θ2Θ=0=0.
p=-2gfp(-T)Θ2Θ=0
=p=-2fp(-T)Θ2Θ=0-4Θw2 fp(-T)(0, Z),
gΘΘ=0=0,2gΘ2Θ=0=-4Θw2
2gfp(-1)Θ2Θ=0-4Θw2+p2Jp(-pζ)+8 xZ pJp(-pζ)-16xZ2Jp(-pζ),
-1<ζ<1,
p=-2gfp(-1)Θ2Θ=0-4Θw211+ζ+ζ[1+ζ]4+8ζΘw21[1+ζ]2-16ζ2Θw40.
2gfp(-2)Θ2Θ=0=1-Θw28Z2 p2 2gfp(-1)Θ2Θ=0,
p=-2gfp(-2)Θ2Θ=0=p=-2gfp(-1)Θ2Θ=0+ζ28x2Θw2-ζ[1+ζ]6+10ζ2[1+ζ]7-ζ22x2Θw4ζ[1+ζ]4.
2gfp(-T)Θ2Θ=0=2gfp(-2)Θ2Θ=0+2Θw2xZ2-p2Jp(-pζ)+8 xZ pJp(-pζ)-16xZ2Jp(-pζ)2gfp(-2)Θ2Θ=0-xZ22Θw2p2Jp(-pζ),
p=-2gfp(-T)Θ2Θ=0p=-2gfp(-2)Θ2Θ=0-4Θw2ζ2[1+ζ]3.
-4Θw2 [1+ζ]3+ζ+8 ζΘw2 [1+ζ]2-4 ζ2[1+ζ]Θw20,
1ζZΘw2xΘw24-1.
z2k¯xh2[h2/w02-1].
g(Θ, Z)p=-fp(-T)(Θ, Z)
exp-2Θ2Θw2p=-exp(ipΘ)×1-p2Θw28Z2Jp-4 p+i4Θ/Θw2Θw2-4+32Θw2[Θw2-4]2 Jp-4 p+i4Θ/Θw2Θw2-4,
Zz/z01,
Θwπ;
hπw0/2,
zz0xπ2[π2/4-1]=45.5×w02xn¯/λ,
|ϕ˘(y, z)|2p=-fp(-0)(Θ, Z)=s=-fs(+0)(Θ, Z)p=-fp(-1)(Θ, Z)=s=-fs(+1)(Θ, Z)p=-fp(-2)(Θ, Z)p=-fp(-T)(Θ, Z)s=-fs(+2)(Θ, Z),
fp(-2)(Θ, Z)exp-p2Θw28Z2+ipΘ×Jp-4x sinp Θw24Z+iΘZ,
fs(+2)(Θ, Z)exp-s2Θw28Z2-s ΘZ×Js-4x sins Θw24Z+Θ.
2p2h2/w2p2Θw2/2Z2|Θ|
2s2h2/w2s2Θw2/2Z2|Θ/Z|
s=-fs(+1)(Θ, Z)=p=-fp(-1)(Θ, Z),
fp(-1)(Θ, Z)exp(ipΘ)Jp-4x sinp Θw24Z+iΘZ,
fs(+1)(Θ, Z)exp(-sΘ/Z)×Js-4x sins Θw24Z+Θ.
fp(-2)(Θ, Z)exp-p2Θw28Z2+ipΘ×Jp-[pΘw2+i4Θ] xZ
fs(+2)(Θ, Z)?exp-s2Θw28Z2-s ΘZ×Js-4xs Θw24Z+Θ
s=-fs(+0)(Θ, Z)=p=-fp(-0)(Θ, Z),
fs(+0)(Θ, Z)Js-4x sins Θw24Z+Θ,
fp(-0)(Θ, Z)=exp(ipΘ)Jp-4x sinp Θw24Z.
p=-fp(-0)(Θ, Z)p=-exp(ipΘ)Jp(-pζ)=11+ζ cos ψ(-ζ, Θ),
ζΘw2x/Z,
ψ(-ζ, Θ)+ζ sin ψ(-ζ, Θ)=Θ
k=-Ck+ν(u)Jk(v)exp(ikα)=Cν(w)exp(iνχ),
w=[u2+v2-2uv cos α]1/2,
sin χ=v sin αw,cos χ=u-v cos αw.
k=-Ck+ν/2(u)Jk-ν/2(v)exp(i[k-ν2 ]α)=Cν(w)exp(iνx),
k=-Ck+ν/2(u)Jk-ν/2(v)exp(ikα)=Cν(w)exp(iν[x+α/2]).
w=u2-2 cos α=2u sin(α/2),
χ=[π-α]/2.
k=-Jk+n/2(u)Jk-n/2(u)exp(ikα)=i±nJ±n(2usin(α/2)).
1=k=-Jk2(z)=J02(z)+2k=1Jk2(z).
k=-ikJk+n/2(u)Jk-n/2(u)exp(ikα)=i±nJ ±n(2usinα2 )ucosα2,
-k=-k2Jk+n/2(u)Jk-n/2(u)exp(ikα)=i±n[J ±n(2usinα2)u2×cos2α2-J ±n(2usinα2)u2sinα2].
1+2n=1Jn(nz)n=-Jn(nz)=11-zn=0zn,
z exp1-z21+1-z2<1.
Jν(z exp(in π))=exp(in νπ)Jν(z),
C-n(z)=[-1]nCn(z),
n=-npdpJn(u)dupu=nz=p![1-z]p+1.
B(z, θ)n=-Jn(nz)exp(in θ)=11-z cos ψ(z, θ)
ψ(z, θ)-z sin ψ(z, θ)=θ.
ψθ=11-z cos ψ(z, θ)=B(z, θ).
θ B(z, θ)=-z sin ψ(z, θ)B2(z, θ)ψθ,
n=-inJn(nz)exp(in θ)=-z sin ψ(z, θ)B3(z, θ),
2θ2 B(z, θ)=-z cos ψ(z, θ)B3(z, θ)ψθ-3z sin ψ(z, θ)B2(z, θ)Bθ
-n=-n2Jn(nz)exp(in θ)=-zB4cos ψ+3z2B5sin2 ψ,
-in=-n3Jn(nz)exp(in θ)=zB5sin ψ+5z2B6sin(2ψ)-15z3B7sin3 ψ,
n=-n4Jn(nz)exp(in θ)
=zB6cos ψ+z2B7[10 cos2 ψ-15 sin2 ψ]-105z3B8sin2 ψ cos ψ-105z4B9sin4 ψ,
n=-n2m+1Jn(nz)=0
-n=-n2Jn(nz)=-zB4(z, 0)=-z[1-z]4,
n=-n4Jn(nz)=zB6(z, 0)+10z2B7(z, 0)=z[1-z]6+10z2[1-z]7,
n=-n3Jn(nz)=ddz {zB4(z, 0)}=1[1-z]4+4z[1-z]5,
ddz B(z, 0)=B2(z, 0)
n=-np+sdpJn(u)dupu=nz=dpdzpn=-nsJn(nz),
n=-n4Jn(nz)=ddz {zB4(z, 0)}=8[1-z]5+20z[1-z]6,
n=1Jn(nz)n=z+24.
n=--1Jn(nz)n=z+24.
n=-n0Jn(nz)n=z2+1.
n=-n0Jn(nz)=12.
J0(0)=-J1(0)=-J0(0)-J2(0)2=-12,
n=-Jn(nz)=0.
s=-exp(sα)Js(κ sin(sβ+Θ))
=p=-exp(ipΘ)Jp(κ sin(pβ-iα))
s=1sinh(sα)Js(κ sin(sβ+Θ))
=p=1sin(pΘ)Jp(κ sin(pβ-iα)),
[-1]p2pθp11-z cos ψ(z, θ)0=0=n=1n2pJn(nz).

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