Abstract

For an efficient analysis of the coupling characteristic of a multimode fiber device using a GRIN-rod lens, a skew ray path in radial gradient suitable for an investigation of the third-order property of a GRIN-rod lens is derived by greatly modifying the first-order asymptotic solution found by Paxton and Streifer. Then, using the derived skew ray path, a coupling characteristic is analyzed on the multimode fiber directional coupler developed by Minemura et al. The result is shown as a contour map of coupling efficiency when the influences of defocus and the third-order aberrations of the GRIN-rod lens and of the lateral shifts of the optical fiber are taken into account. The validity of the ray trace analysis is also examined from the point of view of the far-field pattern of a multimode fiber.

© 1986 Optical Society of America

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References

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  1. W. J. Tomlinson, “Applications of GRIN-Rod Lenses in Optical Fiber Communication Systems,” Appl. Opt. 19, 1127(1980).
    [CrossRef] [PubMed]
  2. K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
    [CrossRef]
  3. K. Thyagarajan, A. Rohra, A. K. Ghatak, “Aberration Losses of the Microoptic Directional Coupler,” Appl. Opt. 19, 1061 (1980).
    [CrossRef] [PubMed]
  4. J. C. Palais, “Fiber Coupling using Graded-Index Lenses,” Appl. Opt. 19, 2011 (1980).
    [CrossRef] [PubMed]
  5. A. Nicia, “Lens Coupling in Fiber-Optic Devices: Efficiency Limits,” Appl. Opt. 20, 3136 (1981).
    [CrossRef] [PubMed]
  6. B. D. Metcalf, L. Jou, “Dual GRIN Lens Wavelength Multiplexer,” Appl. Opt. 22, 455 (1983).
    [CrossRef] [PubMed]
  7. K. B. Paxton, W. Streifer, “Analytic Solution of Ray Equations in Cylindrically Inhomogeneous Guiding Media. 2: Skew Rays,” Appl. Opt. 10, 1164 (1971).
    [CrossRef] [PubMed]
  8. T. W. Cline, R. B. Jander, “Wave-Front Aberration Measurements on GRIN-Rod Lenses,” Appl. Opt. 21, 1035 (1982).
    [CrossRef] [PubMed]
  9. T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
    [CrossRef] [PubMed]
  10. T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).
  11. E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficientsto Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [CrossRef]
  12. Y. Daido, E. Miyauchi, T. Iwama, “Measuring Fiber Connection Loss Using Steady-State Power Distribution: a Method,” Appl. Opt. 20, 451 (1981).
    [CrossRef] [PubMed]
  13. K. B. Paxton, W. Streifer, “Aberrations and Design of Graded-Index (GRIN) Rods used as Image Relays,” Appl. Opt. 10, 2090 (1971).
    [CrossRef] [PubMed]

1985

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

1983

1982

1981

1980

1978

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

1973

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficientsto Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

1971

Cline, T. W.

Daido, Y.

Ghatak, A. K.

Ishikawa, R.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Iwama, T.

Jander, R. B.

Jou, L.

Kobayashi, K.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Metcalf, B. D.

Minemura, K.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Miyauchi, E.

Murray, R. G.

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficientsto Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Nicia, A.

Palais, J. C.

Paxton, K. B.

Rawson, E. G.

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficientsto Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Rohra, A.

Sakamoto, T.

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

T. Sakamoto, “GRIN Lens Profile Measurement by Ray Trace Analysis,” Appl. Opt. 22, 3064 (1983).
[CrossRef] [PubMed]

Shikada, M.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Streifer, W.

Sugimoto, S.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Thyagarajan, K.

Tomlinson, W. J.

Ueki, A.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Yanase, T.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

Appl. Opt.

Electron. Lett.

K. Minemura, K. Kobayashi, T. Yanase, R. Ishikawa, M. Shikada, A. Ueki, S. Sugimoto, “Two-Way Transmission Experiments over a Single Optical Fibre at the Same Wavelength using Micro-Optic 3dB Couplers,” Electron. Lett. 14, 340 (1978).
[CrossRef]

IEEE J. Quantum Electron.

E. G. Rawson, R. G. Murray, “Interferometric Measurement of Selfoc Dielectric Constant Coefficientsto Sixth Order,” IEEE J. Quantum Electron. QE-9, 1114 (1973).
[CrossRef]

Trans. IECE Jpn.

T. Sakamoto, “Accuracy of Optical Path Lengths Computed with an Asymptotic Solution for a Meridional Ray Equation in GRIN-Rod Lenses,” Trans. IECE Jpn. J68-C, 519 (1985).

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Figures (6)

Fig. 1
Fig. 1

Model for the coupling loss analysis of the directional coupler. GRIN-rod lens (Selfoc SLW 1.8-mm lens): n0 = 1.60, g = 0.341 mm−1, lens length Z = Π/g. Optical fibers (graded-index type): N.A.0 = 0.30, 2r0 = 100 μm, off-axis distance x i = 100 um.

Fig. 2
Fig. 2

Histogram of the rays launched from the source fiber: (a) N r = 11; (b) N r = 101; (c) N r = 1001. The ordinate is the normalized number of rays falling in the intervals which are obtained by dividing |N.A.| ≤ 0.3 into 11 equally. Ns and N r denote the numbers of point sources and of the rays launched from the point source at center, respectively. α = 2.0.

Fig. 3
Fig. 3

Influence of α parameter: N s = 101 and N r = 1001.

Fig. 4
Fig. 4

Convergence of the solution for h4 = 0. The abscissa is the total number of rays launched from the source fiber. α = 2.0.

Fig. 5
Fig. 5

Influence of defocus. α = 2.0.

Fig. 6
Fig. 6

Influence of lateral shifts of the receiving fiber: (a) lateral shift Δx; (b) lateral shift Δy. α = 2.0.

Equations (18)

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n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 + h 4 ( g r ) 4 + ] ,
g x = { g x i + h 4 2 4 [ ( g x i ) 3 - 3 g x i x ˙ i 2 + g x i ( g y i ) 2 - g x i y ˙ i 2 - 2 x ˙ i g y i y ˙ i ] - h 4 2 ( - x ˙ i g y i + g x i y ˙ i ) ( g x i g y i + x ˙ i y ˙ i ) ( g x i ) 2 + x ˙ i 2 g x i g z } cos Ω x z + { x ˙ i - x ˙ i 2 [ ( g x i ) 2 + ( g y i ) 2 + x ˙ i 2 + y ˙ i 2 ] + h 4 2 4 [ 9 x ˙ i 3 + 21 ( g x i ) 2 x ˙ i + 7 x ˙ i ( g y i ) 2 + 14 g x i g y i y ˙ i + 9 x ˙ i y ˙ i 2 ] - h 4 2 ( - x ˙ i g y i + g x i y ˙ i ) ( g x i g y i + x ˙ i y ˙ i ) ( g x i ) 2 + x ˙ i 2 x ˙ i g z } sin Ω x z - h 4 2 4 [ g x i ( g y i ) 2 - g x i y ˙ i 2 - 2 x ˙ i g y i y ˙ i ] cos ( Ω x + 2 Ω y ) z - h 4 2 4 [ x ˙ i ( g y i ) 2 - x ˙ i y ˙ i 2 - 2 g x i g y i y ˙ i ] sin ( Ω x + 2 Ω y ) z - h 4 2 4 [ ( g x i ) 3 - 3 g x i x ˙ i 2 ] cos 3 Ω x z - h 4 2 4 [ 3 ( g x i ) 2 x ˙ i - x ˙ i 3 ] sin 3 Ω x z ,
Ω x g = 1 - 3 4 ( h 4 - 2 3 ) [ ( g x i ) 2 + ( g y i ) 2 + x ˙ i 2 + y ˙ i 2 ] + h 4 2 ( - x ˙ i g y i + g x i y ˙ i ) 2 ( g x i ) 2 + x ˙ i 2 ,
N . A . 2 ( r ) = N . A . 0 2 [ 1 - ( r r 0 ) α ] ,
N . A . 2 = n 2 - n i 2 cos 2 γ i ,
η = 10 log 10 N received N launched ,
x = a cos ψ - δ h 4 2 4 [ b 2 a cos ( ψ + 2 ϕ ) + a 3 cos 3 ψ ] , y = b cos ϕ - δ h 4 2 4 [ a 2 b cos ( ϕ + 2 ψ ) + b 3 cos 3 ϕ ] ,
x = ξ r 0 ,             y = η r 0 ,             z = δ 1 / 2 n 0 ζ r 0 n i cos γ i .
a ( z ) = a i - δ h 4 2 z a i b i 2 sin θ i cos θ i , b ( z ) = b i - δ h 4 2 z b i a i 2 sin θ i cos θ i , ψ ( z ) = ψ i + ψ x z , ϕ ( z ) = ϕ i + ϕ x z ,
θ i = ψ i - ϕ i , ρ i 2 = a i 2 + b i 2 , ψ x = 1 - δ h 4 4 ( 3 ρ i 2 - 2 b i 2 sin 2 θ i ) , ϕ y = 1 - δ h 4 4 ( 3 ρ i 2 - 2 a i 2 sin 2 θ i ) ,
a i = a 0 + δ a 1 ,             ψ i = ψ 0 + δ ψ 1 , b i = b 0 + δ b 1 ,     ϕ i = ϕ 0 + δ ϕ 1 .
x = [ a 0 cos ψ 0 - δ ( ψ 1 a 0 sin ψ 0 - a 1 * cos ψ 0 ) ] cos ψ x z - [ a 0 sin ψ 0 + δ ( ψ 1 a 0 cos ψ 0 + a 1 * sin ψ 0 ) ] sin ψ x z - δ h 4 2 4 b 0 2 a 0 cos ( ψ 0 2 ϕ 0 ) cos ( ψ x + 2 ϕ y ) z + δ h 4 2 4 b 0 2 a 0 sin ( ψ 0 2 ϕ 0 ) sin ( ψ x + 2 ϕ y ) z + δ h 4 2 4 a 0 3 cos ψ 0 cos 3 ψ x z - δ h 4 2 4 a 0 3 sin 3 ψ 0 sin 3 ψ x z ,
x i = a 0 cos ψ 0 , y i = b 0 cos ϕ 0 , x ˙ i = - a 0 sin ψ 0 , y ˙ i = - b 0 sin ϕ 0 .
a 1 = h 4 2 4 a 0 ( x i 4 + 18 x i 2 x ˙ i 2 + 9 x ˙ i 4 + x i 2 y i 2 - x i 2 y ˙ i 2 ) + 7 x ˙ i 2 y i 2 + 9 x ˙ i 2 y ˙ i 2 + 12 x i x ˙ i y i y ˙ i ) , b 1 = h 4 2 4 b 0 ( y i 4 + 18 y i 2 y ˙ i 2 + 9 y ˙ i 4 + 7 x i 2 y i 2 - 7 x ˙ i 2 y i 2 + x i 2 y ˙ i 2 + 15 x ˙ i 2 y ˙ i 2 + 36 x i x ˙ i y i y ˙ i ) , ψ 1 = h 4 2 4 a 0 2 ( - 18 x i 2 y i y ˙ i + 2 x ˙ i 2 y i y ˙ i - 2 x i x ˙ i y i 2 - 14 x i x ˙ i y ˙ i 2 - 20 x i 3 x ˙ i - 12 x i x ˙ i 3 ) , ϕ 1 = h 4 2 4 b 0 2 ( - 20 y i 3 y ˙ i - 12 y i y ˙ i 3 - 18 y i 2 x i x ˙ i + 2 y ˙ i 2 x i x ˙ i - 2 y i y ˙ i x i 2 - 14 y i y ˙ i x ˙ i 2 ) .
ψ x z = δ 1 / 2 n 0 ψ x r 0 n i cos γ i ζ .
r 0 Ω x δ 1 / 2 = n 0 ψ x n i cos γ i .
Ω x g = 1 - δ 3 4 ( h 4 - 2 3 ) ( a 0 2 + b 0 2 ) + δ h 4 2 b 0 2 sin 2 θ 0 .
r 0 x ˙ = r 0 n i cos γ i δ 1 / 2 n 0 d ξ d ζ , r 0 y ˙ = r 0 n i cos γ i δ 1 / 2 n 0 d η d ζ .

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