Abstract

On the basis of the overlap integral method, an approximate analytical model is derived to estimate the coupled optical power between axisymmetric Gaussian beams when transverse, axial, and angular misalignments simultaneously exist in three dimensions. Seven optical properties are derived from a detailed analysis of the model. Because the model is an approximate analytical solution to the overlap integral method, the existence of each property is also investigated by a numerical solution. Results show that all seven properties are intrinsic to the optical coupling phenomenon between Gaussian beams. Because numerous single-mode device-to-fiber coupling systems can be well described by use of Gaussian beams, the seven properties provide a solid basis to develop model-based algorithms for single-mode device-to-fiber alignment automation.

© 2004 Optical Society of America

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References

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  1. S. Kidd, “Automated alignment equipment in test and measurement,” Lightwave 18, 80–85 (2001).
  2. S. R. Kidd, C. Buckberry, “All the right moves: developing a successful fiber optic alignment tool requires careful choices about bearings and translation stage designs,” Photonics Spectra 35, 122–125 (2001).
  3. S. Jordan, “Automated fiber alignment pays off for manufacturers,” Laser Focus World 35, 141–144 (1999).
  4. K. Mobarhan, “Aligning fibers to devices demands precision,” WDM Solutions 3, 51–58 (2001).
  5. R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.
  6. Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
    [CrossRef]
  7. W. Gawronski, E. M. Craparo, “Three scanning techniques for deep space network antennas to estimate spacecraft position,” in The Interplanetary Network Progress Report (2001), pp. 42–147, 1–17, http://tmo.jpl.nasa.gov .
  8. J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
    [CrossRef]
  9. H. Kogelnik, “Coupling and conversion coefficients for optical modes in quasi-optics,” in Microwave Research Institute Symposia Series (Polytechnic, New York, 1964), Vol. 14, pp. 333–347.
  10. E.-G. Neumann, Single Mode Fibers I: Fundamentals, Vol. 57 of the Springer Series in Optical Sciences (Springer-Verlag, New York, 1988).
    [CrossRef]
  11. H. Kartensen, “Laser diode to single-mode fiber coupling will ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
  12. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703–719 (1977).
    [CrossRef]
  13. L. A. Wang, C. D. Su, “Tolerance analysis of aligning an astigmatic laser diode with a single-mode optical fiber,” J. Lightwave Technol. 14, 2757–2762 (1996).
    [CrossRef]
  14. W. B. Joyce, B. C. DeLoach, “Alignment of Gaussian beams,” Appl. Opt. 23, 4187–4196 (1984).
    [CrossRef] [PubMed]
  15. S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers using a Gaussian field approximation,” Opt. Quantum Electron. 11, 447–457 (1979).
    [CrossRef]
  16. Y. St-Amant, D. Rancourt, D. Gariépy, “Using fundamental properties of optical coupling for single mode fiber transverse and longitudinal alignment automation,” in Applications of Photonic Technology 6, R. A. Lessard, G. A. Lampropoulos, eds., Proc. SPIE5260, 191–202 (2003).
    [CrossRef]
  17. Y. St-Amant, “Alignement automatisé de fibres optiques amorces monomodes,” Ph.D. dissertation (Mechanical Engineering Department, Université Laval, Québec, Canada, 2004).
  18. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  19. B. E. A. Saleh, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]

2001 (4)

S. Kidd, “Automated alignment equipment in test and measurement,” Lightwave 18, 80–85 (2001).

S. R. Kidd, C. Buckberry, “All the right moves: developing a successful fiber optic alignment tool requires careful choices about bearings and translation stage designs,” Photonics Spectra 35, 122–125 (2001).

K. Mobarhan, “Aligning fibers to devices demands precision,” WDM Solutions 3, 51–58 (2001).

Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
[CrossRef]

1999 (1)

S. Jordan, “Automated fiber alignment pays off for manufacturers,” Laser Focus World 35, 141–144 (1999).

1996 (1)

L. A. Wang, C. D. Su, “Tolerance analysis of aligning an astigmatic laser diode with a single-mode optical fiber,” J. Lightwave Technol. 14, 2757–2762 (1996).
[CrossRef]

1988 (1)

H. Kartensen, “Laser diode to single-mode fiber coupling will ball lenses,” J. Opt. Commun. 9, 42–49 (1988).

1984 (1)

1979 (1)

S. Nemoto, T. Makimoto, “Analysis of splice loss in single-mode fibers 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]

1973 (1)

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Buckberry, C.

S. R. Kidd, C. Buckberry, “All the right moves: developing a successful fiber optic alignment tool requires careful choices about bearings and translation stage designs,” Photonics Spectra 35, 122–125 (2001).

Cook, J. S.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

DeLoach, B. C.

Gariépy, D.

Y. St-Amant, D. Rancourt, D. Gariépy, “Using fundamental properties of optical coupling for single mode fiber transverse and longitudinal alignment automation,” in Applications of Photonic Technology 6, R. A. Lessard, G. A. Lampropoulos, eds., Proc. SPIE5260, 191–202 (2003).
[CrossRef]

Grow, R. J.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Jordan, S.

S. Jordan, “Automated fiber alignment pays off for manufacturers,” Laser Focus World 35, 141–144 (1999).

Joyce, W. B.

Kartensen, H.

H. Kartensen, “Laser diode to single-mode fiber coupling will ball lenses,” J. Opt. Commun. 9, 42–49 (1988).

Kidd, S.

S. Kidd, “Automated alignment equipment in test and measurement,” Lightwave 18, 80–85 (2001).

Kidd, S. R.

S. R. Kidd, C. Buckberry, “All the right moves: developing a successful fiber optic alignment tool requires careful choices about bearings and translation stage designs,” Photonics Spectra 35, 122–125 (2001).

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes in quasi-optics,” in Microwave Research Institute Symposia Series (Polytechnic, New York, 1964), Vol. 14, pp. 333–347.

Makimoto, T.

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

Mammel, W. L.

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

Marcuse, D.

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

Mobarhan, K.

K. Mobarhan, “Aligning fibers to devices demands precision,” WDM Solutions 3, 51–58 (2001).

Mondal, S. K.

R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.

Nemoto, S.

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

Neumann, E.-G.

E.-G. Neumann, Single Mode Fibers I: Fundamentals, Vol. 57 of the Springer Series in Optical Sciences (Springer-Verlag, New York, 1988).
[CrossRef]

Rancourt, D.

Y. St-Amant, D. Rancourt, D. Gariépy, “Using fundamental properties of optical coupling for single mode fiber transverse and longitudinal alignment automation,” in Applications of Photonic Technology 6, R. A. Lessard, G. A. Lampropoulos, eds., Proc. SPIE5260, 191–202 (2003).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Shi, F. G.

Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
[CrossRef]

R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

St-Amant, Y.

Y. St-Amant, D. Rancourt, D. Gariépy, “Using fundamental properties of optical coupling for single mode fiber transverse and longitudinal alignment automation,” in Applications of Photonic Technology 6, R. A. Lessard, G. A. Lampropoulos, eds., Proc. SPIE5260, 191–202 (2003).
[CrossRef]

Y. St-Amant, “Alignement automatisé de fibres optiques amorces monomodes,” Ph.D. dissertation (Mechanical Engineering Department, Université Laval, Québec, Canada, 2004).

Su, C. D.

L. A. Wang, C. D. Su, “Tolerance analysis of aligning an astigmatic laser diode with a single-mode optical fiber,” J. Lightwave Technol. 14, 2757–2762 (1996).
[CrossRef]

Tang, Z.

Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
[CrossRef]

R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.

Wang, L. A.

L. A. Wang, C. D. Su, “Tolerance analysis of aligning an astigmatic laser diode with a single-mode optical fiber,” J. Lightwave Technol. 14, 2757–2762 (1996).
[CrossRef]

Zhang, R.

Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
[CrossRef]

R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

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

J. S. Cook, W. L. Mammel, R. J. Grow, “Effect of misalignments on coupling efficiency of single-mode optical fiber butt joints,” Bell Syst. Tech. J. 52, 1439–1448 (1973).
[CrossRef]

J. Lightwave Technol. (1)

L. A. Wang, C. D. Su, “Tolerance analysis of aligning an astigmatic laser diode with a single-mode optical fiber,” J. Lightwave Technol. 14, 2757–2762 (1996).
[CrossRef]

J. Opt. Commun. (1)

H. Kartensen, “Laser diode to single-mode fiber coupling will ball lenses,” J. Opt. Commun. 9, 42–49 (1988).

Laser Focus World (1)

S. Jordan, “Automated fiber alignment pays off for manufacturers,” Laser Focus World 35, 141–144 (1999).

Lightwave (1)

S. Kidd, “Automated alignment equipment in test and measurement,” Lightwave 18, 80–85 (2001).

Opt. Commun. (1)

Z. Tang, R. Zhang, F. G. Shi, “Effects of angular misalignments on fiber-optic alignment automation,” Opt. Commun. 196, 173–180 (2001).
[CrossRef]

Opt. Quantum Electron. (1)

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

Photonics Spectra (1)

S. R. Kidd, C. Buckberry, “All the right moves: developing a successful fiber optic alignment tool requires careful choices about bearings and translation stage designs,” Photonics Spectra 35, 122–125 (2001).

WDM Solutions (1)

K. Mobarhan, “Aligning fibers to devices demands precision,” WDM Solutions 3, 51–58 (2001).

Other (8)

R. Zhang, S. K. Mondal, Z. Tang, F. G. Shi, “Fiber-optic angular alignment automation: recent progress,” presented at the Technical Program for SMTA Conference on Optoelectronics and the Telecom Revolution, Dallas, Texas, 14–15 Nov. 2001.

W. Gawronski, E. M. Craparo, “Three scanning techniques for deep space network antennas to estimate spacecraft position,” in The Interplanetary Network Progress Report (2001), pp. 42–147, 1–17, http://tmo.jpl.nasa.gov .

H. Kogelnik, “Coupling and conversion coefficients for optical modes in quasi-optics,” in Microwave Research Institute Symposia Series (Polytechnic, New York, 1964), Vol. 14, pp. 333–347.

E.-G. Neumann, Single Mode Fibers I: Fundamentals, Vol. 57 of the Springer Series in Optical Sciences (Springer-Verlag, New York, 1988).
[CrossRef]

Y. St-Amant, D. Rancourt, D. Gariépy, “Using fundamental properties of optical coupling for single mode fiber transverse and longitudinal alignment automation,” in Applications of Photonic Technology 6, R. A. Lessard, G. A. Lampropoulos, eds., Proc. SPIE5260, 191–202 (2003).
[CrossRef]

Y. St-Amant, “Alignement automatisé de fibres optiques amorces monomodes,” Ph.D. dissertation (Mechanical Engineering Department, Université Laval, Québec, Canada, 2004).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

B. E. A. Saleh, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

(a) Three-dimensional, (b) top, and (c) side views of a single-mode device-to-fiber system.

Fig. 2
Fig. 2

Definition of a transverse misalignment line.

Fig. 3
Fig. 3

Optimal transverse misalignment line.

Fig. 4
Fig. 4

Optimal angular misalignment.

Fig. 5
Fig. 5

(a) Raw and (b) normalized coupled optical power along different optimal transverse misalignment lines.

Fig. 6
Fig. 6

Absolute rate of variation of the coupled optical power along all optimal transverse misalignment lines.

Fig. 7
Fig. 7

(a) Two-dimension transverse scans for z d = 500 μm and β d = 0.2 rad. In (b), (c), and (d), simulated coupled optical power (solid circles) is compared with the best paraboloid fit (solid curve) for the three one-dimensional transverse scans highlighted in (a).

Fig. 8
Fig. 8

Numerical investigation of property A: MAVR as a function of longitudinal misalignment for angular misalignments varying from 0 to 0.2 rad.

Fig. 9
Fig. 9

Numerical investigation of property B: MAVR as a function of axial misalignment for θ d = 0 rad (solid curve), θ d = 0.1 rad (dashed curve), and θ d = 0.2 rad (dotted curve). Curves overlay each other for z d > 200 μm.

Fig. 10
Fig. 10

Optimal transverse misalignment: comparison between analytical prediction (solid curve) and numerical results (filled circles).

Fig. 11
Fig. 11

Optimal angular misalignment: comparison between analytical prediction (dotted curve) and numerical results (solid curve).

Fig. 12
Fig. 12

Hyperbolic behavior of the width of the coupled optical power transverse distribution. Comparison between analytical prediction (solid curve) and numerical results for optimal transverse misalignment line corresponding to β d = 0 rad (dashed curve), β d = 0.1 rad (dotted curve), and β d = 0.2 rad (dashed-dotted curve).

Fig. 13
Fig. 13

(a) Raw and (b) normalized simulated coupled optical power along optimal transverse misalignment lines corresponding to β d = 0 rad (solid curve), β d = 0.05 rad (dashed curve), β d = 0.1 rad (dotted curve), and β d = 0.2 rad (dashed-dotted curve). The analytical prediction (heavy solid curve) is superimposed over the numerical simulation for β d = 0 rad.

Fig. 14
Fig. 14

Absolute rate of variation of simulated coupled optical power along optimal transverse misalignment lines corresponding to β d = 0 rad (solid curve), β d = 0.05 rad (dashed curve), β d = 0.1 rad (dotted curve), and β d = 0.2 rad (dashed-dotted curve). The analytical prediction (heavy solid curve) is superimposed over the numerical simulation for β d = 0 rad.

Fig. 15
Fig. 15

Wang and Su’s notation and sign convention: (a) three-dimensional, (b) top, and (c) side views.

Fig. 16
Fig. 16

Nemoto and Makimoto’s notation and sign convention.

Tables (1)

Tables Icon

Table 1 Numerical Investigation of Property G

Equations (56)

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P=-- ψ1x, y, zψ2*x, y, zdxdy2-- ψ2x, y, zψ2*x, y, zdxdy,
P=-- ψ1x, yψ2*x, ydxdy2.
P=PxPy,
Px=4σG2+σ+121/2 exp-pσ+1Fx2-2σFxG sin βd+σG2+σ+1sin2 βdG2+σ+12,
Py=4σG2+σ+121/2 exp-pσ+1Fy2-2σFyG sin αd+σG2+σ+1sin2 αdG2+σ+12,
σ=w02w012, p=2πnw01λ2, Fx=xdzR1, Fy=ydzR1, G=zdzR1, zR1=πnw012λ.
xd=mxzd, yd=myzd,
mx=tan θd, my=tan ϕd.
PdB=10 logP.
xd-xcd2a2+ yd-ycd2b2+ PdB-PcdBc=0,
a=b=z01ln 1010pG2+σ+12σ+11/2, c=1,
xcd= σσ+1 zd sin βd, ycd=- σσ+1 zd sin αd,
PcdB=10 log4σG2+σ+12-10pln 10σσ+1×sin2 αd+sin2 βd.
αd-αcd2a2+ βd-βcd2b2+ PdB-PcdBc=0,
a=b=ln 1010pG2+σ+12σG2+σ+11/2, c=1,
αcd=- FyGG2+σ+1, βcd= FxGG2+σ+1,
PcdB=10 log4σG2+σ+12-10pln 10×Fx2+Fy2G2+σ+1.
xoptd=σσ+1sin βdzd,yoptd=-σσ+1sin αdzd.
ϕoptd=tan-1- σσ+1sin αd,θoptd=tan-1σσ+1sin βd.
ϕoptd=- σσ+1 αd, θoptd= σσ+1 βd.
αoptd=- FyGG2+σ+1, βoptd= FxGG2+σ+1.
αoptd- FyG=-ydzd, βoptd FxG= xdzd.
αoptd-ϕd, βoptdθd.
sd=xd-xcd2+yd-ycd21/2.
sda2-zdb2=1,
a=zR1ΔPdBσ+1ln1010p1/2, b=zR1σ+1,
ΔPdB=PcdB-PdB.
PdB=PcdB =10 log4σG2+σ+12-10pln10σσ+1×sin2 αd+sin2 βd.
PndB=PdB-PG=0dB=10 logσ+12G2+σ+12.
PdBzd=-20ln10GG2+σ+121zR1.
zd=σ+1zR1.
Δx= 2|A||Δy|.
CL= Δx|Δy|= 2|A|.
η=ηxηy,
ηx=ηzxηdxηθxηd,θx,
ηzx= 2wfwx11wf2+ 1wx22+ k241Rx- 1Rf21/2,
ηdx=exp-2dx21wx21wf21wx2+ 1wf2+ k241wx2Rf2+ 1wf2Rx21wx2+ 1wf22+ k241Rx- 1Rf2,
ηθx=exp- k2θy221wx2+ 1wf21wx2+ 1wf22+ k241Rx- 1Rf2,
ηd,θx=expk2θydx1Rfwx2+ 1Rxwf21wx2+ 1wf22+ k241Rx- 1Rf2.
η= 4σG2+σ+12exp-pσ+1F2+2σFG sin θ+σG2+σ+1sin2 θG2+σ+12,
σ=w2w12, p=2πniw1λ2, F= szR1, G= zwzR1, zR1= πniw12λ.
wx=wxz=w0x1+zzR21/2,
Rx=Rxz=z1+zRz2,
η=ηzxηdxηθxηd,θxηzyηdyηθyηd,θy,
ηzx=ηzy=4w0f2w0x21+z/zR21D21/2,
ηdx=exp-2dx21w0x21+z/zR21w0f2×1w0x21+z/zR2+ 1w0f2+ 2π/λm241+ 1w0f2z21+zR/z21D,
ηθx=exp- 2π/λm2θy221w0x21+z/zR2+ 1w0f21D,
ηd,θx=exp-2π/λm2dxθy1R0fw0x21+z/zR2+ 11D,
ηdy=ηd,θy=ηθy=1,
D=1w0f2+ 1w0x21+z/zR22+ 2π/λm241z1+zR/z2- 121/2.
ηzx=ηzy=4σG2+σ+121/2,
ηdx=exp-pσ+1F2G2+σ+12,
ηθx=exp-pσG2+σ+1θy2G2+σ+12,
ηd,θx=exp-p2σFGθyG2+σ+12,
ηdy=ηd,θy=ηθy=1,
p=2πw0xλm2, σ=w0fw0x2, F= dxzR, G= zzR.

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