Abstract

We investigate free-space diffraction of light that emanates from obliquely cleaved end faces of single-mode fibers. Emphasis is placed on precise prediction of the wave-front sphericity of fiber-generating waves in the nonparaxial Fresnel propagation region spanning an entire hemispherical observation surface. Rayleigh–Sommerfeld scalar diffraction theory is used to produce an analytic closed-form solution with a nonparaxial approximation. The result allows the wave front’s sphericity and the amplitude distribution of fiber-generating waves to be evaluated precisely with less computation than for existing numerical or infinite-series solutions.

© 2004 Optical Society of America

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References

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2003

2002

2001

1999

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1999).
[CrossRef]

1998

1996

G. E. Sommargren, Laser Focus World 32, 61 (1996).

1992

1987

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, J. Opt. Soc. Am. A 4, 2150 (1987).
[CrossRef]

1983

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

1982

1980

L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

Belland, P.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1999).
[CrossRef]

Capsalis, C. N.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, J. Opt. Soc. Am. A 4, 2150 (1987).
[CrossRef]

Chanclou, P.

Chen, C. G.

Chen, W. T.

Crenn, J. P.

de Bougrenet de la Tocnaye, J.-L.

Duan, K.

Ferrera, J.

Gradshteyn, L. S.

L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

Heilmann, R. K.

Hrynevych, M.

Kim, S.-W.

Konkola, P. T.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

Lü, B.

Nourrit, V.

Rhee, H.-G.

Ryzhik, I. M.

L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

Schattenburg, M. L.

Sheppard, C. J. R.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

Sommargren, G. E.

G. E. Sommargren, Laser Focus World 32, 61 (1996).

Tigelis, I.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, J. Opt. Soc. Am. A 4, 2150 (1987).
[CrossRef]

Tomlinson, W. J.

Uzunoglu, N. K.

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, J. Opt. Soc. Am. A 4, 2150 (1987).
[CrossRef]

Wager, R. E.

Weng, L. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1999).
[CrossRef]

Appl. Opt.

J. Lightwave Technol.

J. Opt. Soc. Am. A

N. K. Uzunoglu, C. N. Capsalis, and I. Tigelis, J. Opt. Soc. Am. A 4, 2150 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Laser Focus World

G. E. Sommargren, Laser Focus World 32, 61 (1996).

Opt. Lett.

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, New York, 1999).
[CrossRef]

L. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980).

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Figures (4)

Fig. 1
Fig. 1

Spherical coordinates for diffraction analysis of a fiber with oblique angle φ.

Fig. 2
Fig. 2

Fizeau interferometer for a flat test: (a) conventional design with a pinhole and a beam splitter, (b) new design with an oblique single-mode fiber.

Fig. 3
Fig. 3

Normalized amplitude profiles when φ=29.19°, ϕ=0°, w0=1.5 µm, λ=0.633 µm, and R/z0=1000 (i.e., R=11.17 mm): (a) Exact numerical solution of Eq. (2), (b) nonparaxial approximation of relation (3), (c) paraxial Fresnel approximation.

Fig. 4
Fig. 4

Phase distribution with the same approximations as in Fig. 3.

Equations (8)

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ux,y,0=exp-x2 cos2 φw02-y2w02expiknx sin φ=exp-x2w12-y2w22+iknx sin φ,
uR,θ,ϕ=12πAux,y,0expikrr×R cos θrik+1rdxdy.
rR-x cos ϕ+y sin ϕsin θ+x2+y2/2R-x cos ϕ+y sin ϕ2 sin2 θ/2R.
uR,θ,ϕi cos θλRAexp-x2w12-y2w22+iknx sin φexpikR-x cos ϕ sin θ-y×sin ϕ sin θ-xy2Rsin2ϕ sin2 θ+x22R1-cos2×ϕ sin2 θ+y22R1-sin2 ϕ sin2 θdxdy.
uR,θ,ϕ=i cos θαβ+sin2 ϕ cos2 ϕ sin4 θ×expkRi- βδ2+α+iδ cos ϕ sin θsin2 ϕ sin2 θ2αβ+sin2 ϕ cos2 ϕ sin4 θ,
α=R/z1-i1-cos2 ϕ sin2 θ,β=R/z2-i1-sin2 ϕ sin2 θ,δ=cosϕ sin θ-n sin φ,    z1=kw12/2,z2=kw22/2.
uϕ=0=cos θ R/z12+cos4 θ-1/4R/z22+1-1/4×exp-kz12sin θ-n sin φ21+z1/R2 cos4 θ,
uϕ=0=kR-kR2sin θ-n sin φ2 cos2 θR/z12+cos4 θ-12 tan-1Rz1 cos2 θ-12 tan-1Rz2.

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