Abstract

We show that the coupling efficiency from a laser diode (LD) to an optical fiber through a ball lens can be calculated accurately using the exact solution to Maxwell’s equations for the scattering of a beam from a dielectric sphere. Our calculated results agree closely with coupling measurements from an asymmetric LD for two different ball lenses.

© 1997 Optical Society of America

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References

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  1. R. P. Ratowsky, L. Yang, R. J. Deri, J. S. Kallman, G. Trott, “Ball lens reflections by direct solution of Maxwell’s equations,” Opt. Lett. 20, 2048–2050 (1995).
    [CrossRef] [PubMed]
  2. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  3. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957); J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988); G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985).
  4. A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
    [CrossRef]
  5. M. Sumida, K. Takemoto, “Lens coupling of laser diodes to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984); H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988); H. Karstensen, K. Drogemuller, “Loss analysis of laser diode to single-mode fiber couplers with glass spheres or silicon plano-convex lenses,” J. Lightwave Technol. 8, 739–747 (1990); J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
    [CrossRef]
  6. H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
    [CrossRef]
  7. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.
  8. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658–661 (1983).
    [CrossRef] [PubMed]

1995

1993

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

1986

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

1984

M. Sumida, K. Takemoto, “Lens coupling of laser diodes to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984); H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988); H. Karstensen, K. Drogemuller, “Loss analysis of laser diode to single-mode fiber couplers with glass spheres or silicon plano-convex lenses,” J. Lightwave Technol. 8, 739–747 (1990); J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[CrossRef]

1983

1975

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957); J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988); G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985).

Abeles, J. H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

Barber, P. W.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Brodwin, M. E.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Deri, R. J.

Hill, S. C.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.

Kallman, J. S.

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Logan, R. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957); J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988); G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985).

Olsson, N. A.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

Panish, M. B.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

Ratowsky, R. P.

Self, S. A.

Sumida, M.

M. Sumida, K. Takemoto, “Lens coupling of laser diodes to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984); H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988); H. Karstensen, K. Drogemuller, “Loss analysis of laser diode to single-mode fiber couplers with glass spheres or silicon plano-convex lenses,” J. Lightwave Technol. 8, 739–747 (1990); J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[CrossRef]

Taflove, A.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

Takemoto, K.

M. Sumida, K. Takemoto, “Lens coupling of laser diodes to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984); H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988); H. Karstensen, K. Drogemuller, “Loss analysis of laser diode to single-mode fiber couplers with glass spheres or silicon plano-convex lenses,” J. Lightwave Technol. 8, 739–747 (1990); J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[CrossRef]

Temkin, H.

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

Trott, G.

Yang, L.

Ann. Phys. (Leipzig)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidalen Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908); H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957); J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988); G. Gouesbet, G. Grehan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. 16, 83–93 (1985).

Appl. Opt.

IEEE J Quantum Electron

H. Temkin, N. A. Olsson, J. H. Abeles, R. A. Logan, M. B. Panish, “Reflection noise in index-guided InGaAsP lasers,” IEEE J Quantum Electron QE-22, 286–293 (1986).
[CrossRef]

IEEE Trans. Antennas Propag.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. Taflove, M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. MTT-23, 623–630 (1975).
[CrossRef]

J. Lightwave Technol.

M. Sumida, K. Takemoto, “Lens coupling of laser diodes to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984); H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988); H. Karstensen, K. Drogemuller, “Loss analysis of laser diode to single-mode fiber couplers with glass spheres or silicon plano-convex lenses,” J. Lightwave Technol. 8, 739–747 (1990); J. S. Kim, S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. 73, 303–312 (1983).
[CrossRef]

Opt. Lett.

Other

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.

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Figures (6)

Fig. 1
Fig. 1

Scattering geometry for the on-axis case x 0 = y 0 = 0.

Fig. 2
Fig. 2

Calculated coupling efficiency versus SMF lens distance D SMF for seven values of the LD lens distance D LD. The lens is 300-µm diameter sapphire; the LD spot size is 1.02 × 0.713 µm at λ = 1.3 µm; the SMF spot size is 4.5 × 4.5 µm2.

Fig. 3
Fig. 3

Calculated and measured peak coupling efficiency versus SMF lens distance D SMF for 300-µm diameter sapphire and BK-7 ball lenses. Other parameters are as in Fig. 2.

Fig. 4
Fig. 4

Optimal SMF lens distance D SMF (solid curve) and coupling efficiency (dashed curve) versus LD lens distance D LD for 300-µm diameter BK-7 ball lens. The upper dashed curve is calculated with Eq. (5).

Fig. 5
Fig. 5

Calculated and measured peak coupling efficiency versus LD offset (x 0) from peak coupling point, for sapphire and BK-7 ball lenses. The LD is translated in the x direction. Other parameters are as in Fig. 2.

Fig. 6
Fig. 6

Calculated intensity (solid curves) and phase (dotted curves) versus x in the plane of peak coupling for (a) 0-µm, (b) 5-µm, and (c) 10-µm LD offsets. Each curve is labeled by D SMF, the SMF lens distance at which lineout is taken. The intensity scale is normalized to unit incident beam peak intensity.

Equations (7)

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Exx, y; z=-z0=E0 exp-x-x02/σx2-y-y02/σy2
Eout=Einc+Escat.
Evr, θ, ϕ=l=1il4π2l+1m=-1lalvflvkvr×ul,m++blvkv-1×flvkvrul,m-  v=inc, scat, int
u1m±θ, ϕ=r×Yl,m+1θ, ϕ±Yl,m-1θ, ϕ;
CE=Einc*·Einc+Escatdxdy2Einc2dxdy,
DSMF+R=f+DLD+R-fDLD+R-f2+zR2,
CE=4T2M2σSMF2σxσyσSMF2+M2σx2σf2+M2σSMF2,

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