Abstract

We show that the coupling efficiency of an optical system to a waveguide can be related to an apodized and normalized point spread function. Adapting the problem to the use of optical design software, we analyze and optimize systems that contain a fair amount of aberration. We compare theoretical predictions with experimental results and obtain good agreement.

© 1996 Optical Society of America

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References

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  1. M. Sumida, K. Takemoto, “Lens coupling of laser diode to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984).
    [CrossRef]
  2. H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
    [CrossRef]
  3. R. E. Wagner, W. J. Tomlinson, “Coupling efficiency of optics in single-mode fiber components,” Appl. Opt. 21, 2671–2688 (1982).
    [CrossRef] [PubMed]
  4. S. Szapiel, J. Côté, “Analysis of coupling efficiency in a single-mode-fiber component as an optical design problem,” in Optical Design for Photonics Technical Digest, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 140–142.
  5. H. Kogelnik, “Coupling and conversion coefficients for optical mode in quasi-optics,” Microwave Res. Inst. Symp. Ser. 14, 333– 347 (1964).
  6. B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
    [CrossRef]
  7. B. Hillerich, “Shape analysis and coupling loss of microlenses on single-mode fiber tips,” Appl. Opt. 27, 3102 (1988).
    [CrossRef] [PubMed]
  8. M. Côté, “Optimization of waveguide coupling lenses using optical design software,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1995).

1989 (1)

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

1988 (2)

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

B. Hillerich, “Shape analysis and coupling loss of microlenses on single-mode fiber tips,” Appl. Opt. 27, 3102 (1988).
[CrossRef] [PubMed]

1984 (1)

M. Sumida, K. Takemoto, “Lens coupling of laser diode to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984).
[CrossRef]

1982 (1)

1964 (1)

H. Kogelnik, “Coupling and conversion coefficients for optical mode in quasi-optics,” Microwave Res. Inst. Symp. Ser. 14, 333– 347 (1964).

Côté, J.

S. Szapiel, J. Côté, “Analysis of coupling efficiency in a single-mode-fiber component as an optical design problem,” in Optical Design for Photonics Technical Digest, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 140–142.

Côté, M.

M. Côté, “Optimization of waveguide coupling lenses using optical design software,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1995).

Hillerich, B.

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

B. Hillerich, “Shape analysis and coupling loss of microlenses on single-mode fiber tips,” Appl. Opt. 27, 3102 (1988).
[CrossRef] [PubMed]

Karstensen, H.

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical mode in quasi-optics,” Microwave Res. Inst. Symp. Ser. 14, 333– 347 (1964).

Sumida, M.

M. Sumida, K. Takemoto, “Lens coupling of laser diode to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984).
[CrossRef]

Szapiel, S.

S. Szapiel, J. Côté, “Analysis of coupling efficiency in a single-mode-fiber component as an optical design problem,” in Optical Design for Photonics Technical Digest, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 140–142.

Takemoto, K.

M. Sumida, K. Takemoto, “Lens coupling of laser diode to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984).
[CrossRef]

Tomlinson, W. J.

Wagner, R. E.

Appl. Opt. (2)

J. Lightwave Technol. (2)

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

M. Sumida, K. Takemoto, “Lens coupling of laser diode to single-mode fibers,” J. Lightwave Technol. LT-2, 305–311 (1984).
[CrossRef]

J. Opt. Commun. (1)

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Microwave Res. Inst. Symp. Ser. (1)

H. Kogelnik, “Coupling and conversion coefficients for optical mode in quasi-optics,” Microwave Res. Inst. Symp. Ser. 14, 333– 347 (1964).

Other (2)

S. Szapiel, J. Côté, “Analysis of coupling efficiency in a single-mode-fiber component as an optical design problem,” in Optical Design for Photonics Technical Digest, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 140–142.

M. Côté, “Optimization of waveguide coupling lenses using optical design software,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1995).

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Figures (5)

Fig. 1
Fig. 1

General layout of a lens coupling system. The expressions Ψ L and Ψ F represent the laser diode and the fiber and waveguide modal fields described in the entrance pupil (EP) and exit pupil (XP). P is the pupil function of the coupling optics.

Fig. 2
Fig. 2

Ball lens coupling assembly.

Fig. 3
Fig. 3

Coupling efficiency degradation that results from fiber shift along the optical axis.

Fig. 4
Fig. 4

Coupling efficiency degradation that results from lens shift perpendicular to the optical axis.

Fig. 5
Fig. 5

Coupling efficiency degradation that results from fiber shift perpendicular to the optical axis.

Tables (1)

Tables Icon

Table 1 Comparison of the Experimental Data Taken from Ref. 1 with the Calculated Values of the Apodized Scaled Ratios for Five Different Coupling Ball Lenses a

Equations (16)

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η = | ψ L ( x , y ) ψ F * ( x , y ) d x d y | 2 ψ L ( x , y ) ψ L * ( x , y ) d x d y ψ F ( x , y ) ψ F * ( x , y ) d x d y ,
η = | ψ L ( x , y ) P ( x , y ) ψ F * ( x , y ) d x d y | 2 ψ L ( x , y ) P ( x , y ) ψ L * ( x , y ) P * ( x , y ) d x d y ψ F ( x , y ) ψ F * ( x , y ) d x d y ,
P ( x , y ) ψ L ( x , y ) = T ( x , y ) exp [ i k W ( x , y ) ] ψ L ( x , y ) = ψ L ( x , y ) exp [ i k W ( x , y ) ] ,
η = | ψ L ( x , y ) T ( x , y ) exp [ i k W ( x , y ) ] ψ F * ( x , y ) d x d y | 2 ψ L ( x , y ) T ( x , y ) ψ L * ( x , y ) T * ( x , y ) d x d y ψ F ( x , y ) ψ F * ( x , y ) d x d y | ψ L ( x , y ) T ( x , y ) ψ F * ( x , y ) d x d y | 2 | ψ L ( x , y ) T ( x , y ) ψ F * ( x , y ) d x d y | 2 .
A ( x , y ) = ψ L ( x , y ) ψ F * ( x , y ) = ψ L ( x , y ) T ( x , y ) ψ F * ( x , y ) ,
η = | ψ L ( x , y ) ψ F * ( x , y ) d x d y | 2 ψ L ( x , y ) ψ L * ( x , y ) d x d y ψ F ( x , y ) ψ F * ( x , y ) d x d y | A ( x , y ) exp [ i k W ( x , y ) ] d x d y | 2 | A ( x , y ) d x d y | 2 , η = η paraxial ( Ψ L , ψ F ) S R ( A , W ) .
η accepted = ψ L ( x , y ) T ( x , y ) ψ L * ( x , y ) T * ( x , y ) d x d y ψ L ( x , y ) ψ L * ( x , y ) d x d y .
η energy = η η accepted .
Δ z = 8 ( F / # ) 2 W 020 ,
( y + Δ y ) 2 y 2 + 2 y Δ y ,
ψ F * ( x , y + Δ y ) ψ F * ( x , y ) exp ( 2 y Δ y w F 2 ) exp ( iky Δ y R F ) ψ F * ( x , y ) exp ( iky Δ y R F ) ,
S R [ A , W ( x , y + Δ y ) ] = | A ( x , y ) exp [ ikW ( x , y ) ] exp ( 2 π i y Δ y λ R F ) d x d y | 2 | A ( x , y ) d x d y | 2 .
α = Δ y λ R F , β = 0 ,
S R [ A , W ( x , y + Δ y ) ] = | FT [ A ( x , y ) exp [ ikW ( x , y ) ] ] α = Δ y λ R F , β = 0 | 2 | A ( x , y ) d x d y | 2 = PSF ( Δ y ) | A ( x , y ) d x d y | 2 .
η accepted = 1 exp ( 2 y E 2 w LEP 2 ) ,
y E w LEP = 2 π w O L λ n 1 n m ( m 1 ) ,

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