Abstract

A general formula for determining the coupling loss between two single-mode fiber collimators with the simultaneous existence of separation, lateral offset and angular tilt misalignments, and spot-size mismatch is theoretically derived by use of the Gaussian field approximation. Based on this general formula, the formulas for coupling losses that are due to the misalignment of insert separation, lateral offset, and angular tilt are given. The formula for the coupling loss that is due to Gaussian spot-size mismatch of two single-mode collimators is also given. Good agreement between these formulas and experimental results is demonstrated with gradient-index rod lens-based fiber collimators operating in the 1300-nm band.

© 1999 Optical Society of America

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References

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  1. M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
    [CrossRef]
  2. W. J. Tomlinson, “Application of GRIN-rod lenses in optical fiber communication systems,” Appl. Opt. 19, 1127–1138 (1980).
    [CrossRef] [PubMed]
  3. N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
    [CrossRef]
  4. N. Madamopoulos, N. Riza, “Directly modulated semiconductor-laser-fed photonic delay line with ferroelectric liquid crystals,” Appl. Opt. 37, 1407–1416 (1998).
    [CrossRef]
  5. J. C. Palais, “Fiber coupling using graded-index rod lenses,” Appl. Opt. 19, 2011–2018 (1980).
    [CrossRef] [PubMed]
  6. R. W. Gilsdorf, J. C. Palais, “Single-mode fiber coupling efficiency with graded-index rod lenses,” Appl. Opt. 33, 3440–3445 (1994).
    [CrossRef] [PubMed]
  7. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.
  8. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
    [CrossRef]
  9. S. Nemeto, T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
    [CrossRef]
  10. J. K. Kim, N. A. Riza, “Fiber array optical coupling design issues for photonic beamformers,” in Pacific Northwest Fiber Optic Sensor Workshop, E. Udd, ed., Proc. SPIE2754, 271–282 (1996).
  11. A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).
  12. Selfoc Product Guide, manufacturer’s literature on fiber collimators (NSG America, Inc., New Jersey, 1997).
  13. T. Sakamoto, “Coupling characteristic analysis of single-mode and multimode optical-fiber connectors using gradient-index-rod lenses,” Appl. Opt. 31, 5184–5190 (1992).
    [CrossRef] [PubMed]
  14. W. S. Wu, “Reduction of coupling loss in many-to-many collimating system for optomechanical matrix switch,” Opt. Eng. 37, 1834–1837 (1998).
    [CrossRef]
  15. Opto-electronics Group, Corning, Inc., Corning Optical Fiber Product Information PI1044, on Corning SMF-28 single-mode optical fiber (Corning, Inc., Corning, N.Y., 1996).
  16. N. A. Riza, S. Yuan, “Demonstration of a liquid-crystal adaptive alignment tweeker for high-speed infrared band fiber-fed free-space system,” Opt. Eng. 37, 1876–1880 (1998).
    [CrossRef]

1998 (4)

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

W. S. Wu, “Reduction of coupling loss in many-to-many collimating system for optomechanical matrix switch,” Opt. Eng. 37, 1834–1837 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Demonstration of a liquid-crystal adaptive alignment tweeker for high-speed infrared band fiber-fed free-space system,” Opt. Eng. 37, 1876–1880 (1998).
[CrossRef]

N. Madamopoulos, N. Riza, “Directly modulated semiconductor-laser-fed photonic delay line with ferroelectric liquid crystals,” Appl. Opt. 37, 1407–1416 (1998).
[CrossRef]

1997 (1)

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

1994 (1)

1992 (1)

1980 (2)

1979 (1)

S. Nemeto, T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
[CrossRef]

Borella, M. S.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Gilsdorf, R. W.

Jue, J. P.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Kim, J. K.

J. K. Kim, N. A. Riza, “Fiber array optical coupling design issues for photonic beamformers,” in Pacific Northwest Fiber Optic Sensor Workshop, E. Udd, ed., Proc. SPIE2754, 271–282 (1996).

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.

Madamopoulos, N.

Makimoto, T.

S. Nemeto, T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
[CrossRef]

Mukherjee, B.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Nemeto, S.

S. Nemeto, T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Palais, J. C.

Ramamurthy, B.

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Riza, N.

Riza, N. A.

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Demonstration of a liquid-crystal adaptive alignment tweeker for high-speed infrared band fiber-fed free-space system,” Opt. Eng. 37, 1876–1880 (1998).
[CrossRef]

J. K. Kim, N. A. Riza, “Fiber array optical coupling design issues for photonic beamformers,” in Pacific Northwest Fiber Optic Sensor Workshop, E. Udd, ed., Proc. SPIE2754, 271–282 (1996).

Sakamoto, T.

Tomlinson, W. J.

Wu, W. S.

W. S. Wu, “Reduction of coupling loss in many-to-many collimating system for optomechanical matrix switch,” Opt. Eng. 37, 1834–1837 (1998).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

Yuan, S.

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Demonstration of a liquid-crystal adaptive alignment tweeker for high-speed infrared band fiber-fed free-space system,” Opt. Eng. 37, 1876–1880 (1998).
[CrossRef]

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
[CrossRef]

Electron. Lett. (1)

N. A. Riza, S. Yuan, “Low optical interchannel crosstalk, fast switching time, polarization independent 2 × 2 fiber optic switch using ferroelectric liquid crystals,” Electron. Lett. 34, 1341–1342 (1998).
[CrossRef]

Opt. Eng. (2)

W. S. Wu, “Reduction of coupling loss in many-to-many collimating system for optomechanical matrix switch,” Opt. Eng. 37, 1834–1837 (1998).
[CrossRef]

N. A. Riza, S. Yuan, “Demonstration of a liquid-crystal adaptive alignment tweeker for high-speed infrared band fiber-fed free-space system,” Opt. Eng. 37, 1876–1880 (1998).
[CrossRef]

Opt. Quantum Electron. (1)

S. Nemeto, T. Makimoto, “Analysis of splice loss in single-mode fibres using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
[CrossRef]

Proc. IEEE (1)

M. S. Borella, J. P. Jue, B. Ramamurthy, B. Mukherjee, “Components for WDM lightwave networks,” Proc. IEEE 85, 1274–1307 (1997).
[CrossRef]

Other (5)

J. K. Kim, N. A. Riza, “Fiber array optical coupling design issues for photonic beamformers,” in Pacific Northwest Fiber Optic Sensor Workshop, E. Udd, ed., Proc. SPIE2754, 271–282 (1996).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford U. Press, New York, 1997).

Selfoc Product Guide, manufacturer’s literature on fiber collimators (NSG America, Inc., New Jersey, 1997).

Opto-electronics Group, Corning, Inc., Corning Optical Fiber Product Information PI1044, on Corning SMF-28 single-mode optical fiber (Corning, Inc., Corning, N.Y., 1996).

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J. Fox, ed., Vol. 14 of Polytechnic Institute Microwave Research Institute Symposia Series (Polytechnic Brooklyn, Brooklyn, N.Y., 1964), pp. 335–347.

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Figures (14)

Fig. 1
Fig. 1

Fiber coupling system using two quarter-pitch SMF collimators.

Fig. 2
Fig. 2

Gaussian beam propagation in the fiber collimating system.

Fig. 3
Fig. 3

Output optical field distribution from GRIN lens 1.

Fig. 4
Fig. 4

Equivalent optical field distribution that can be perfectly coupled into the SMF2 by use of GRIN lens 2.

Fig. 5
Fig. 5

(a) Separation misalignment between the two GRIN lens surfaces. (b) Offset misalignments between the longitudinal axes of the GRIN lenses. (c) Angular misalignments between the longitudinal axes of the GRIN lenses.

Fig. 6
Fig. 6

Side view of two fiber collimators with three combined misalignments in two rectangular systems [(x, y, z) and (x′, y′, z′)].

Fig. 7
Fig. 7

Changing curves of beam radius w(z) for several Gaussian waists w g .

Fig. 8
Fig. 8

Relationship of divergence angle θ d and Gaussian beam waist w g .

Fig. 9
Fig. 9

Coupling-loss variation owing to changing of Gaussian waist w g .

Fig. 10
Fig. 10

Theoretical result of the normalized coupling loss that is due to the mismatch of the Gaussian beam spot sizes.

Fig. 11
Fig. 11

Measured results of the output power distribution with motion of the blade: (a) 162.1-µm waist for the fiber collimator with SN 464584, (b) 175.9-µm waist for the fiber collimator with SN 464585.

Fig. 12
Fig. 12

Comparison of experimental and theoretical results for the normalized coupling loss that is due to the separation between the two fiber collimators.

Fig. 13
Fig. 13

Comparison of the experimental and theoretical results for the normalized coupling loss that is due to the lateral offset when (a) Z 0 = 0, i.e., butt-coupling situation, and (b) Z 0 = 10 cm.

Fig. 14
Fig. 14

Comparison of experimental and theoretical results for the normalized coupling loss that is due to the angular tilt when (a) Z 0 = 0, i.e., butt-coupling situation and (b) Z 0 = 10 cm.

Equations (53)

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

1qi=1Ri-j λπnwi2,
nr=n01-Ar22,
cosAZ1n0AsinAZ-n0AsinAZcosAZ,
G=A1B1C1D1=01n0A-n0A0.
S=A2B2C2D2=1Z001.
qi+1=Aiqi+BiCiqi+Di.
w1=λπnw0n0A,
R1=,
w2=w11+λZ0πnw1221/2,
R2=Z01+πnw12λZ02,
w3=λn0Aπnw1=w0,
R3=-1n0A2Z0.
Exx, y, z=E1wTwzexp-ikz-ηz-r21w2z+i k2Rz,
k=2πn/λ,
ηz=tan-1λzπnwT2,
w2z=wT21+λzπnwT22,
Rz=z1+πnwT2λz2.
Exx, y, z=E1wTwzexp-ikz-ηz-r21w2z+i k2Rz,
ηz=tan-1λzπnwR2,
w2z=wR21+λzπnwR22,
Rz=z1+πnwR2λz2.
ηc=2πE12wT2-+-+ Exx, y, z|z=0×Ex*x, y, z|z=0dxdy.
x=x cos θ-z sin θ+X0,
z=x sin θ+z cos θ+Z0,
y=y,
r2=x2+y2=x cos θ-z sin θ+X02+y2.
Exx, y, z|z=0=Exx+X0, y, x sin θ+Z0=E1wTwZ0exp-ikZ0-ηZ0×exp-1w2Z0+i k2RZ0 x2+1w2Z0+i k2RZ0×2X0+ik sin θ x+1w2Z0+i k2RZ0X02×exp-1w2Z0+i k2RZ0 y2.
Exx, y, z|z=0=E1wTwRexp-x2wR2exp-y2wR2.
-+exp-ax2+bx+cdx=πaexpb2-4ac4a,
ηc=C0 exp-AC+jH2Bexp-jψ0,
C0=4DB1/2,
ψ0=AG-tan-1GD+1,
A=kwT2/2,
B=G2+D+12,
C=D+1F2+2DFG sin θ+DG2+D+1sin2 θ,
D=wR/wT2,
F=2X0kwT2,
G=2Z0kwT2,
H=GF2-2DD+1F sin θ-GD2 sin θ.
T=4DBexp-ACB.
LtotX0, Z0, θ=-10 log T=-10 log4DBexp-ACB.
Ls=LtotX0=0, Z0, θ=0-LtotX0=0, Z0=0, θ=0=-10 log4wT2wR2λ2Z02π2n2+wT2+wR22.
Ll=LtotX0, Z0, θ=0-LtotX0=0, Z0, θ=0=20ln 10n2π2wT2+wR2λ2Z02+π2n2wT2+wR22 X02.
La=LtotX0=0, Z0, θ-LtotX0=0, Z0, θ=0=20ln 10nπwRλ2λZ0πnwT22+wRwT2+1λZ0πnwT22+wRwT2+12sin2 θ.
Lm=LtotX0=0, Z0=0, θ=0, wT, wR-LtotX0=0, Z0=0, θ=0, wT, wR=wT=-10 log4wRwT+wTwR2.
LsZ0=10log1+λ2Z024π2n2wg4.
Pbx=P021-erf2xw,
L=-10logPm/P0.
Ex, y  1wexp-r2w2=1wexp-x2+y2w2,
Ix, y  |Ex, y|2  1w2exp-2x2+2y2w2.
-- Ix, ydxdy=P0,
Ix, y=2P0πw2exp-2x2+2y2w2.
Pbx0=-x0 Ix, ydxdy=2P0πw2-x0 exp-2x2+2y2w2dxdy=2P0πw2-exp-2y2w2dy x0 exp-2x2w2dx=P0w2πx0 exp-2x2w2dx=P021-erf2x0w,

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