Abstract

Arbitrary modal field coupling is analyzed and optimized by using tapered and nontapered GRIN fiber lenses as single-mode optical fiber connectors. Conditions for achieving maximum coupling efficiency are derived.

© 1990 Optical Society of America

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References

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  1. H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
    [CrossRef]
  2. K. Lizuka, Engineering Optics (Springer-Verlag, Berlin, 1987),. Chap. 13.
  3. E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).
  4. H. R. D. Sunak, “Single-Mode Fiber Measurements,” IEEE Trans. Instrum. Meas. IM-27, 557–560 (1989).
  5. W. L. Emkey, C. A. Jack, “Analysis and Evaluation of Graded-Index Fiber-Lenses,” IEEE/OSA J. Lightwave Technol. LT-5, 1156–1164 (1987).
    [CrossRef]
  6. R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1984), Chap. 6.
  7. C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission Through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A 3, 1604–1607 (1986).
    [CrossRef]
  8. C. Gomez-Reino, J. Linares, “Optical Path Integrals in Graded-Index Media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
    [CrossRef]
  9. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.
  10. C. Gomez-Reino, M. V. Perez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372–375 (1983).
    [CrossRef]
  11. K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
    [CrossRef]

1989 (2)

E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).

H. R. D. Sunak, “Single-Mode Fiber Measurements,” IEEE Trans. Instrum. Meas. IM-27, 557–560 (1989).

1987 (3)

W. L. Emkey, C. A. Jack, “Analysis and Evaluation of Graded-Index Fiber-Lenses,” IEEE/OSA J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
[CrossRef]

C. Gomez-Reino, J. Linares, “Optical Path Integrals in Graded-Index Media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

1986 (2)

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission Through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A 3, 1604–1607 (1986).
[CrossRef]

1983 (1)

C. Gomez-Reino, M. V. Perez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372–375 (1983).
[CrossRef]

Acosta, E.

E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).

Emkey, W. L.

W. L. Emkey, C. A. Jack, “Analysis and Evaluation of Graded-Index Fiber-Lenses,” IEEE/OSA J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

Flores, J. R.

E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).

Gomez-Reino, C.

E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).

C. Gomez-Reino, J. Linares, “Optical Path Integrals in Graded-Index Media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

C. Gomez-Reino, J. Linares, E. Larrea, “Imaging and Transforming Transmission Through Tapered Gradient-Index Rods: Analytical Solutions,” J. Opt. Soc. Am. A 3, 1604–1607 (1986).
[CrossRef]

C. Gomez-Reino, M. V. Perez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372–375 (1983).
[CrossRef]

Honmow, H.

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1984), Chap. 6.

Ishikawa, R.

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

Jack, C. A.

W. L. Emkey, C. A. Jack, “Analysis and Evaluation of Graded-Index Fiber-Lenses,” IEEE/OSA J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

Kobayashi, M.

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

Larrea, E.

Linares, J.

Lizuka, K.

K. Lizuka, Engineering Optics (Springer-Verlag, Berlin, 1987),. Chap. 13.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.

Pal, B. P.

K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
[CrossRef]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372–375 (1983).
[CrossRef]

Sarkar, S. N.

K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
[CrossRef]

Sunak, H. R. D.

H. R. D. Sunak, “Single-Mode Fiber Measurements,” IEEE Trans. Instrum. Meas. IM-27, 557–560 (1989).

Thyagarajan, K.

K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
[CrossRef]

Ueno, H.

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

Electron. Lett. (1)

H. Honmow, R. Ishikawa, H. Ueno, M. Kobayashi, “1.0-dB Low-Loss Coupling of Laser Diode to Single-Mode Fiber Using a Plano-Convex Graded-index Rod Lens,” Electron. Lett. 22, 1122–1123 (1986).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

H. R. D. Sunak, “Single-Mode Fiber Measurements,” IEEE Trans. Instrum. Meas. IM-27, 557–560 (1989).

IEEE/OSA J. Lightwave Technol. (2)

W. L. Emkey, C. A. Jack, “Analysis and Evaluation of Graded-Index Fiber-Lenses,” IEEE/OSA J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

K. Thyagarajan, S. N. Sarkar, B. P. Pal, “Equivalent Step Index (ESI) Model for Elliptic Core Fibers,” IEEE/OSA J. Lightwave Technol. LT-5, 1041–1044 (1987).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Gomez-Reino, M. V. Perez, E. Larrea, “Modal Propagation in Gradient-Index (GRIN) Material,” Opt. Commun. 45, 372–375 (1983).
[CrossRef]

Opt. Eng. (1)

E. Acosta, J. R. Flores, C. Gomez-Reino, J. Linares, “Gradient Index Lens Law for Gaussian Illumination: Image and Focal Shifts,” Opt. Eng. 28, 1168–1172 (1989).

Other (3)

K. Lizuka, Engineering Optics (Springer-Verlag, Berlin, 1987),. Chap. 13.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1984), Chap. 6.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964), Chap. 4.

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Figures (2)

Fig. 1
Fig. 1

Arrangement for tapered GFLs fused to the end of single-mode fibers.

Fig. 2
Fig. 2

Distance z between nontapered GFLs vs gradient parameter Calculations have been made for g1 = 5mm−1, n1 = n2 = 1.5, and A = 0.9.

Equations (61)

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η = | - ψ n 1 ( α x , α y ) ψ n 2 ( x , y ) d x d y | 2 ,
ψ n 1 ( α x , α y ) = ψ 1 ( α x , α y ) ψ 1 ( α x , α y ) ψ 1 * ( α x , α y ) α 2 d x d y ,
ψ n 2 ( x , y ) = ψ 2 ( x , y ) ψ 2 ( x , y ) ψ 2 * ( x , y ) d x d y ,
d η / d α α = α m = 0 ;
α m = σ o / ( ω x ω y ) 1 / 2 ;
σ o = ( ω x ω y ) 1 / 2 .
n ( x , y , z ) = n o [ 1 - g 2 ( z ) 2 ( x 2 + y 2 ) ] ,
G ( x , y , x o , y o , z ) = F ( z ) exp ( i k S ( x , y , z , x o , y o ) = k / 2 π i exp [ - ½ 2 ( p x x - p x o x o ) d z ] × exp [ - ½ 2 ( p y y - p y 0 y o ) d z ] × exp [ i k / 2 ( p x x - p x o x o ) ] exp [ i k / 2 ( p y y - p y o y o ) ] ,
S = n o z + ½ ( p x x + p y y ) o .
q 1 = A q 0 + B p q o ,
p q 1 = C q 0 + D p q 0 ,
ψ n 1 ( x , y ) = R I 2 G ( x , y , x o , y o , z ) ψ n 1 ( x o , y o ) d x o d y o .
G ( x , y , x 0 , y 0 , z ) = 1 / A exp { i k C 2 A [ ( x 2 + y 2 ) ] } × δ ( x - x 0 A , y - y 0 A ) .
ψ n 1 ( x , y ) = 1 / A exp { i k C 2 A [ ( x 2 + y 2 ) ] } ψ n 1 ( x / A , y / A ) .
B ( z 1 , , z n ) = 0 ,
C ( z 1 , , z n ) = 0 ,
M k = [ S 2 ( k ) S 1 ( k ) / n k n k S 2 ( k ) S 1 ( k ) ] ,
M T = M 2 [ 1 z 0 1 ] M 1 = [ A B C D ] ,
A = S 2 ( 2 ) [ S 2 ( 1 ) + n 1 z S ˙ 2 ( 1 ) ] + ( n 2 / n 1 ) S 1 ( 2 ) S ˙ 2 ( 1 ) ,
B = S 2 ( 2 ) [ S 1 ( 1 ) / n 1 + z S ˙ 1 ( 1 ) ] + S 1 ( 2 ) S ˙ 1 ( 1 ) / n 2 ,
C = n 2 S ˙ 2 ( 2 ) [ S 2 ( 1 ) + n 1 z S ˙ 2 ( 1 ) ] + n 1 S ˙ 1 ( 2 ) S ˙ 2 ( 1 ) ,
D = n 2 S ˙ 2 ( 2 ) [ S 1 ( 1 ) / n 1 + z S ˙ 1 ( 1 ) ] + S ˙ 1 ( 2 ) S ˙ 1 ( 1 ) .
S ˙ 1 ( 1 ) = S ˙ 2 ( 2 ) = 0 ,
z = - [ S ˙ 1 ( 2 ) / n 2 S ˙ 2 ( 2 ) + s 2 ( 1 ) / n 1 S ˙ 2 ( 1 ) ] .
M = [ ( n 1 / n 2 ) S 1 ( 2 ) S ˙ 2 ( 1 ) 0 0 ( n 2 / n 1 ) S ˙ 2 ( 2 ) S 1 ( 1 ) ] .
α = n 2 / n 1 S ˙ 1 ( 2 ) S 2 ( 1 ) .
n 1 S ˙ 1 ( 2 ) S ˙ 2 ( 1 ) = - n 2 S ˙ 2 ( 2 ) S 2 ( 1 ) ,
n 2 S 1 ( 1 ) S 2 ( 2 ) = - n 1 S 1 ( 2 ) S ˙ 1 ( 1 ) ,
α = S 2 ( 2 ) S 2 ( 1 ) + ( n 2 / n 1 ) S 1 ( 2 ) S ˙ 2 ( 1 ) ,
S 2 ( k ) = S ˙ 1 ( k ) = 0 ;
α = - ( n 2 / n 1 ) S 1 ( 2 ) / S 1 ( 1 ) .
S 1 ( k ) = sin [ g ( z ) z ] / [ g o k g k ( z ) ] 1 / 2 ,
S 2 ( k ) = g o k 1 / 2 cos [ g ( z ) z ] / [ g k ( z ) ] 1 / 2 ,
g k ( z n ) z n = π ( n + ½ ) ,
α = ± ( n 2 / n 1 ) [ g 1 ( z n ) g 1 o g 2 ( z n ) g 2 o ] 1 / 2 .
S 1 ( k ) = sin ( g k z k ) / g k ,             S 2 ( k ) = cos ( g k z k ) .
cos β [ cos γ - n 1 g 1 z sin γ ] - ( n 1 g 1 / n 2 g 2 ) sin γ sin β = A ,
cos β [ sin γ / n 1 g 1 + z cos γ ] + cos γ sin β / n 2 g 2 = 0 ,
n 2 g 2 sin β [ cos γ - n 1 g 1 z sin γ ] + n 1 g 1 sin γ cos β = 0 ,
z = - tan β / n 2 g 2 - tan γ / n 1 g 1 ,
z = cot β / n 2 g 2 + cot γ / n 1 g 1 .
sin γ cos γ sin β cos β = - u ,
u = n 2 g 2 n 1 g 1 .
cos β / cos γ = A = 1 / α .
cos γ = ± ( 1 - A 2 u 2 ) ( 1 - A 4 u 2 ) ,
sin γ = ± u A ( 1 - A 2 ) ( 1 - A 4 u 2 )
cos β = ± A ( 1 - A 2 u 2 ) ( 1 - A 4 u 2 ) ,
sin β = ± ( 1 - A 2 ) ( 1 - A 4 u 2 ) .
0 < 1 - A 2 u 2 1 - A 4 u 2 < 1.
A < 1 or equivalently α > 1 ,
u [ 0 , 1 / A ] = [ 0 , α ] .
z = ( 1 / n 2 g 2 A ) 1 - A 2 u 2 1 - A 2 { ( ± A 2 + 1 ) for tan γ > 0 , ( ± A 2 - 1 ) for tan γ < 0.
1 = ω x 1 / ω y 1 ( ω x 1 > ω y 1 ) ;
2 = ω x 2 / ω y 2 ( ω x 2 > ω y 2 ) ;
ψ n 1 ( x , y ) = ( ω x 1 ω y 1 π ) - 1 / 2 exp ( - x 2 / 2 ω x 1 2 - y 2 / 2 ω y 1 2 ,
ψ n 2 ( x , y ) = ( ω x 2 ω y 2 π ) - 1 / 2 exp ( - x 2 / 2 ω x 2 2 - y 2 / 2 ω y 2 2 ) .
η ( α ) = 4 ω x 1 ω x 2 ω y 1 ω y 2 α 2 ( ω x 1 2 + α 2 ω x 2 2 ) ( ω y 1 2 + α 2 ω y 2 2 ) ,
α m = [ ω x 1 ω y 1 ω x 2 ω y 2 ] 1 / 2 .
η ( α = α m ) = 4 ω x 1 ω y 1 [ ω x 1 2 + ω x 1 ω x 2 ω y 1 / ω y 2 ] [ ω y 1 2 + ω x 1 ω y 2 ω y 1 / ω x 2 ] ;
[ ω x 1 ω y 1 ω x 2 ω y 2 ] 1 / 2 = cos ( g 1 z 1 ) cos ( g 2 z 2 ) ,
n 2 g 2 / n 1 g 1 = [ ω x 1 ω y 1 ω x 2 ω y 2 ] 1 / 2 .

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