Abstract

We discuss the efficiency with which coherent plane waves can be coupled to single-mode fibers in the presence of deterministic or stochastic misalignments of the fiber relative to the focal point of a lens. We point out how the alignment demands can be relaxed by means of graded-index-lens fiber-pigtailed collimators.

© 2002 Optical Society of America

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References

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  3. K. A. Winick, P. Kumar, “Spatial mode matching efficiencies for heterodyned GaAlAs semiconductor lasers,” J. Lightwave Technol. 6, 513–520 (1988).
    [CrossRef]
  4. K. Tanaka, N. Ohta, “Effects of tilt and offset of signal field on heterodyne efficiency,” Appl. Opt. 26, 627–632 (1987).
    [CrossRef] [PubMed]
  5. S. Yuan, N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
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  6. P. J. Winzer, W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23, 986–988 (1998).
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    [CrossRef]
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    [CrossRef]
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  14. K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1987).
    [CrossRef]
  15. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).
  16. E.-G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
    [CrossRef]
  17. O. Wallner, “Kopplung ebener Wellen in Monomode-Glasfasern,” Diploma Thesis (Vienna University of Technology, Vienna, Austria, 2000).
  18. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

1999 (1)

1998 (1)

1995 (1)

D. K. Jacob, M. B. Mark, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

1991 (1)

1988 (1)

K. A. Winick, P. Kumar, “Spatial mode matching efficiencies for heterodyned GaAlAs semiconductor lasers,” J. Lightwave Technol. 6, 513–520 (1988).
[CrossRef]

1987 (1)

1984 (1)

1982 (1)

1978 (1)

1975 (1)

1966 (1)

Buck, J. A.

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).

Cohen, S. C.

Duncan, B. D.

D. K. Jacob, M. B. Mark, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Frehlich, R. G.

Fukumitsu, O.

Iizuka, K.

K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1987).
[CrossRef]

Jacob, D. K.

D. K. Jacob, M. B. Mark, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Kavaya, M. J.

Kumar, P.

K. A. Winick, P. Kumar, “Spatial mode matching efficiencies for heterodyned GaAlAs semiconductor lasers,” J. Lightwave Technol. 6, 513–520 (1988).
[CrossRef]

Leeb, W. R.

Mark, M. B.

D. K. Jacob, M. B. Mark, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Neumann, E.-G.

E.-G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
[CrossRef]

Ohta, N.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

Riza, N. A.

Ruilier, C.

C. Ruilier, “A study of degraded light coupling into single-mode fibers,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 319–329 (1998).
[CrossRef]

Siegman, A. E.

Spiers, G. D.

G. D. Spiers, “The effect of aberrations on the performance of a coherent Doppler lidar,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mt. Hood, Oregon, 28 June–2 July 1999.

Takenaka, T.

Tanaka, K.

Tomlinson, W. J.

Wagner, R. E.

Wallner, O.

O. Wallner, “Kopplung ebener Wellen in Monomode-Glasfasern,” Diploma Thesis (Vienna University of Technology, Vienna, Austria, 2000).

Wang, J. Y.

Winick, K. A.

K. A. Winick, P. Kumar, “Spatial mode matching efficiencies for heterodyned GaAlAs semiconductor lasers,” J. Lightwave Technol. 6, 513–520 (1988).
[CrossRef]

Winzer, P. J.

Yuan, S.

Appl. Opt. (8)

J. Lightwave Technol. (1)

K. A. Winick, P. Kumar, “Spatial mode matching efficiencies for heterodyned GaAlAs semiconductor lasers,” J. Lightwave Technol. 6, 513–520 (1988).
[CrossRef]

Opt. Eng. (1)

D. K. Jacob, M. B. Mark, B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Opt. Lett. (1)

Other (7)

C. Ruilier, “A study of degraded light coupling into single-mode fibers,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 319–329 (1998).
[CrossRef]

G. D. Spiers, “The effect of aberrations on the performance of a coherent Doppler lidar,” presented at the Tenth Biennial Coherent Laser Radar Technology and Applications Conference, Mt. Hood, Oregon, 28 June–2 July 1999.

K. Iizuka, Engineering Optics (Springer-Verlag, Berlin, 1987).
[CrossRef]

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995).

E.-G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988).
[CrossRef]

O. Wallner, “Kopplung ebener Wellen in Monomode-Glasfasern,” Diploma Thesis (Vienna University of Technology, Vienna, Austria, 2000).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

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Figures (7)

Fig. 1
Fig. 1

Coupling geometry. A thin lens of focal length f and clear aperture radius a focuses the incident field EA(x, y) onto the bare end of a single-mode fiber, which generates a field E(x, y) in coupling plane ℬ at z = f + Δz. The field that results from backpropagation of the fiber’s fundamental mode F(x, y) (mode-field radius w) into aperture plane A is designated FA(x, y).

Fig. 2
Fig. 2

Coupling efficiency η normalized to the maximum value of η0 = 0.81 in the case of deterministic misalignment for optimal (χ0) and suboptimal (χ = 0.7χ0) design parameter χ: (a) for normalized lateral misalignment Δx′ = Δx/w and (b) for normalized tilt Δφx′ = Δφx/(a/f).

Fig. 3
Fig. 3

Contour lines of constant normalized coupling efficiency η/η0 as a function of normalized lateral misalignment Δx′ = Δx/w and normalized tilt Δφx′ = Δφx/(a/f) for deterministic parameters. The solid curves represent optimum design parameter χ = χ0 and the dashed curves represent χ = 0.7χ0.

Fig. 4
Fig. 4

Mean coupling efficiency 〈η〉 normalized to the maximum value η0 = 0.81 for stochastic misalignment for optimal χ0 (solid curve) and suboptimal χ = 0.7χ0 (dashed curve) design parameter χ: (a) for normalized lateral misalignment Δx′ = Δx/w with standard deviation σΔx and (b) for normalized tilt Δφx = Δφx/(a/f) with standard deviation σΔφx.

Fig. 5
Fig. 5

Contour lines of constant normalized mean coupling efficiency 〈η〉/η0 as a function of normalized lateral misalignment Δx′ = Δx/w with standard deviation σΔx and normalized tilt Δφx′ = Δφx/(a/f) with standard deviation sfgr;Δφx as stochastic parameters. The solid curves represent optimum design parameter χ = χ0 and the dashed curves represent χ = 0.7χ0.

Fig. 6
Fig. 6

Coupling geometry with a pigtailed collimator. A quarter-pitch GRIN lens is attached directly to the fiber end. It effectively increases the mode-field radius from w to wC.

Fig. 7
Fig. 7

Curves of constant mean coupling efficiency 〈η〉 = 0.75η0 as a function of the standard deviations of normalized lateral misalignment σΔx/w and normalized tilt σΔφx/(a/f) = σΔφxπw/(λχ) for several magnification factors A.

Tables (1)

Tables Icon

Table 1 Normalized Misalignment Parameters

Equations (12)

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η=A EA*(x, y)FA(x, y) dxdy2A |EA(x, y)|2dxdy,
FA(x, y)=2πwA2exp-1wA2-j πΔzλf2×x-Δφxf2+y-Δφyf2×expj2πΔxλfx-Δφx f+Δyλfy-Δφy f,
η=2πχμ2+ν2χ2exp-1-j Δzχ×μ-χΔφx2+ν-χΔφy2×expj2Δxμ+Δyνdμdν2,
η=2χ21-exp-χ22.
η=2πχ2Q˜Q˜exp-μ2+ν2+μ˜2+ν˜2+j Δzχμ2+ν2-μ˜2-ν˜2×exp2χ2μ+μ˜-jΔzχμ-μ˜2σΔφx21+4χ2σΔφx21+4χ2σΔφx21/2×exp2χ2ν+ν˜-jΔzχν-ν˜2σΔφy21+4χ2σΔφy21+4χ2σΔφy21/2×exp-2μ-μ˜2σΔx2-2ν-ν˜2σΔy2dμdνdμ˜dν˜,
pΔx(Δx)=12πσΔxexp-12ΔxσΔx2.
η= ηΔx, Δy, Δφx, Δφy, Δz×pΔx,Δy,Δφx,ΔφyΔx, Δy, Δφx, Δφy×dΔx dΔydΔφxdΔφy,
pΔx,Δy, Δφx,ΔφyΔx, Δy, Δφx, Δφy=pΔxΔx pΔyΔypΔφxΔφx pΔφyΔφy.
η=2πχμ2+ν2χ2exp-1-j Δzχ×μ-χΔφx2+ν-χΔφy2×expj2Δxμ+Δyνdμdν2.
η=2πχ2QQ˜exp-μ2+ν2+μ˜2+ν˜2+j Δzχμ2+ν2-μ˜2-ν˜2×exp-2χ2Δφx2-μ+μ˜-j Δzχ×μ-μ˜1χΔφxΔφx×exp-2χ2Δφy2-ν+ν˜-j Δzχ×ν-ν˜1χ ΔφyΔφy×expj2μ-μ˜ΔxΔx×expj2(ν-ν˜)ΔyΔydμdνdμ˜dν.
exp-bax+x2x=expa2b2σx2/2+4bσx2/1+2bσx2,
expjaxx=exp(-a2σx2/2,

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