Abstract

A new form is proposed for the ABCD matrix in a graded index taper with a large variation in cross section such as might be used for single-mode beam expansion. Expressions are given for the loss of power from the fundamental mode and the coupling efficiency between fibers when two tapers are used in an expanded beam connector. Exact solutions are found for linear tapers and for a class of tapers with zero slope ends. The distinction between adiabatic and nonadiabatic tapers is made clear from the functional form of the matrix in the linear case. Comparisons are made with previously published results and the effect of taper shape on the coupling efficiency is discussed.

© 1989 Optical Society of America

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  1. N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
    [CrossRef]
  2. H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
    [CrossRef]
  3. F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
    [CrossRef]
  4. H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
    [CrossRef]
  5. D. Marcuse, “Mode Conversion in Optical Fibers with Monoton ically Increasing Core Radius,” IEEE/OSA J. Lightwave Tech nol. LT-5, 125 (1987).
    [CrossRef]
  6. C. P. Botham, “Theory of Tapering Single-Mode Optical Fibres by Controlled Core Diffusion,” Electron. Lett. 24, 243 (1988).
    [CrossRef]
  7. J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
    [CrossRef]
  8. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  9. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984), pp. 128–129.
  10. S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Proc. IEEE 34, 1254 (1971).
    [CrossRef]
  11. J. N. McMullin, “The ABCD Matrix in Arbitrarily Tapered Quadratic-Index Waveguides,” Appl. Opt. 25, 2184 (1986).
    [CrossRef] [PubMed]
  12. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Wave guides (Plenum, New York, 1977).
    [CrossRef]
  13. L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
    [CrossRef]
  14. W. J. Stewart, J. D. Love, “Design Limitation on Tapers and Couplers in Single-Mode Fibres,” in Technical Digest, Fifth International Conference on Integrated Optics and Optical Fiber Communication–Eleventh European Conference on Optical Communication, Venice (1985), pp. 559–562.
  15. D. Bertilone, A. Ankiewicz, C. Pask, “Wave Propagation in a Graded-Index Taper,” Appl. Opt. 26, 2213 (1987).
    [CrossRef] [PubMed]
  16. H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
    [CrossRef]
  17. H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1965), p. 105.

1988 (5)

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
[CrossRef]

C. P. Botham, “Theory of Tapering Single-Mode Optical Fibres by Controlled Core Diffusion,” Electron. Lett. 24, 243 (1988).
[CrossRef]

J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
[CrossRef]

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

1987 (4)

D. Marcuse, “Mode Conversion in Optical Fibers with Monoton ically Increasing Core Radius,” IEEE/OSA J. Lightwave Tech nol. LT-5, 125 (1987).
[CrossRef]

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

D. Bertilone, A. Ankiewicz, C. Pask, “Wave Propagation in a Graded-Index Taper,” Appl. Opt. 26, 2213 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

1971 (1)

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Proc. IEEE 34, 1254 (1971).
[CrossRef]

Amitay, N.

H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
[CrossRef]

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

Ankiewicz, A.

Benner, A.

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
[CrossRef]

Bertilone, D.

Botham, C. P.

J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
[CrossRef]

C. P. Botham, “Theory of Tapering Single-Mode Optical Fibres by Controlled Core Diffusion,” Electron. Lett. 24, 243 (1988).
[CrossRef]

Casperson, L. W.

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

DiMarcello, F. V.

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1965), p. 105.

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Wave guides (Plenum, New York, 1977).
[CrossRef]

Harper, J. S.

J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984), pp. 128–129.

Hornung, S.

J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
[CrossRef]

Hussey, C. D.

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

Love, J. D.

W. J. Stewart, J. D. Love, “Design Limitation on Tapers and Couplers in Single-Mode Fibres,” in Technical Digest, Fifth International Conference on Integrated Optics and Optical Fiber Communication–Eleventh European Conference on Optical Communication, Venice (1985), pp. 559–562.

Makimoto, T.

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Proc. IEEE 34, 1254 (1971).
[CrossRef]

Marcuse, D.

D. Marcuse, “Mode Conversion in Optical Fibers with Monoton ically Increasing Core Radius,” IEEE/OSA J. Lightwave Tech nol. LT-5, 125 (1987).
[CrossRef]

Martinez, F.

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

McMullin, J. N.

Nelson, K. T.

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

Pask, C.

Payne, F. P.

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

Presby, H. M.

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
[CrossRef]

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

Scotti, R.

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Wave guides (Plenum, New York, 1977).
[CrossRef]

Stewart, W. J.

W. J. Stewart, J. D. Love, “Design Limitation on Tapers and Couplers in Single-Mode Fibres,” in Technical Digest, Fifth International Conference on Integrated Optics and Optical Fiber Communication–Eleventh European Conference on Optical Communication, Venice (1985), pp. 559–562.

Wylangowski, G.

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

Yamamoto, S.

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Proc. IEEE 34, 1254 (1971).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

Appl. Opt. (2)

Electron. Lett. (5)

H. M. Presby, N. Amitay, R. Scotti, A. Benner, “Simplified Laser to Fibre Coupling via Optical Fibre Up-tapers,” Electron. Lett. 24, 323 (1988)
[CrossRef]

F. Martinez, G. Wylangowski, C. D. Hussey, F. P. Payne, “Practical Single-Mode Fibre-Horn Beam Expander,” Electron. Lett. 24, 14 (1988).
[CrossRef]

H. M. Presby, N. Amitay, A. Benner, “Straight-Tip Optical Fibre Up-Tapers for Single-Mode Hardware Applications,” Electron. Lett. 24, 34 (1988).
[CrossRef]

C. P. Botham, “Theory of Tapering Single-Mode Optical Fibres by Controlled Core Diffusion,” Electron. Lett. 24, 243 (1988).
[CrossRef]

J. S. Harper, C. P. Botham, S. Hornung, “Tapers in Single Mode Optical Fibre by Controlled Core Diffusion,” Electron. Lett. 24, 244 (1988).
[CrossRef]

IEEE/OSA J. Lightwave Tech nol. (1)

D. Marcuse, “Mode Conversion in Optical Fibers with Monoton ically Increasing Core Radius,” IEEE/OSA J. Lightwave Tech nol. LT-5, 125 (1987).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (3)

N. Amitay, H. M. Presby, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers—A Novel Approach to Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 70 (1987).
[CrossRef]

H. M. Presby, N. Amitay, F. V. DiMarcello, K. T. Nelson, “Optical Fiber Tapers at 1.3 μm for Self-Aligned Beam Expansion and Single-Mode Hardware,” IEEE/OSA J. Lightwave Technol. LT-5, 1123 (1987).
[CrossRef]

L. W. Casperson, “Beam Propagation in Tapered Quadratic Index Waveguides: Analytical Solutions,” IEEE/OSA J. Lightwave Technol. LT-3, 264 (1985).
[CrossRef]

Proc. IEEE (1)

S. Yamamoto, T. Makimoto, “Equivalence Relations in a Class of Distributed Optical Systems—Lenslike Media,” Proc. IEEE 34, 1254 (1971).
[CrossRef]

Other (5)

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Wave guides (Plenum, New York, 1977).
[CrossRef]

W. J. Stewart, J. D. Love, “Design Limitation on Tapers and Couplers in Single-Mode Fibres,” in Technical Digest, Fifth International Conference on Integrated Optics and Optical Fiber Communication–Eleventh European Conference on Optical Communication, Venice (1985), pp. 559–562.

H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1965), p. 105.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ, 1984), pp. 128–129.

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Figures (5)

Fig. 1
Fig. 1

Loss of mode 0 in decibels for single linear and raised cosine graded index tapers. Parameters: a1 = 4 μm, a2 = 361 μm, Δ = 0.0056.

Fig. 2
Fig. 2

Coupling loss in decibels for linear and raised cosine graded index expanded beam couplers. Parameters: a1 = 4 μm, a2 = 48 μm, Δ = 0.0056. Irregularities at large L are due to having an insufficient number of points.

Fig. 3
Fig. 3

Shapes of 12 tapers with three different radii compared with raised cosine tapers with aI = 182.5 μm, 1 = 0.01.

Fig. 4
Fig. 4

Shapes of 12 tapers with three different 1 values and a1 = 182.5 μm compared with raised cosine tapers.

Fig. 5
Fig. 5

Coupling loss in decibels for 12 and raised cosine expanded beam couplers. Parameters: a1 = 4 μm, a2 = 361 μm, Δ = 0.0056.

Equations (74)

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[ A B C D ] = [ [ Q ( 0 ) / Q ( z ) ] 1 / 2 α ( z ) [ Q ( 0 ) Q ( z ) ] 1 / 2 β ( z ) [ Q ( 0 ) Q ( z ) ] 1 / 2 γ ( z ) [ Q ( 0 ) / Q ( z ) ] 1 / 2 δ ( z ) ] ,
n 2 ( r , z ) = n 1 2 [ 1 Q 2 ( z ) r 2 ] .
Q ( z ) = 2 Δ a ( z ) .
d A / d z = C ( z ) ,
d B / d z = D ( z ) ,
d C / d z = Q 2 ( z ) A ( z )  ,
d D / d z = Q 2 ( z ) B ( z )  .
t ( z ) = 0 z Q ( z ) d z = 0 z 2 Δ d z a ( z )  ,
d α / d t = [ Ω ( t ) α γ ] ;
d β / d t = [ Ω ( t ) β + δ ] ;
d γ / d t = [ Ω ( t ) γ + α ] ;
d δ / d t = [ Ω ( t ) δ β ] .
Ω ( t ) = 1 2 [ d d t  ln Q ( z ) ] = 1 2 [ d d t  ln a ( z ) ] = ( 8 Δ ) 1 / 2 d a d z .
AD BC = α δ + β γ = 1    .
η ( z ) = | 2 π 0 E a ( r , z ) E b * ( r , z ) r d r | 2 .
E ( r , z ) = 1 q ( z ) { k π  Im [ q ( z ) ] } 1 / 2  exp  [ j k r 2 2 q ( z ) ] .
η ( z ) = 4 [ Im q ( a ) ( z ) ] [ Im q ( b ) ( z ) ] | q ( a ) ( z ) [ q ( b ) ( z ) ] | 2 .
q 0 ( a ) = j a ( 0 ) / ( 2 Δ ) 1 / 2 = j / Q ( 0 ) = j / Q 0 .
q L ( a ) = A q 0 ( a ) +  B C q 0 ( a ) +  D .
q L ( b ) = j / Q ( L ) = j / Q L ,
η L = 4 [ 2 + α L 2 + β L 2 + γ L 2 + δ L 2 ]    .
[ D B C A ] = [ [ Q 0 / Q L ] 1 / 2 δ L [ Q 0 Q L ] 1 / 2 β L [ Q 0 Q L ] 1 / 2 γ L [ Q 0 / Q L ] 1 / 2 ]
[ A B C D ] = [ D B C A ] [ A B C D ] = [ ( α L δ L β L γ L ) 1 Q 0 ( 2 β L δ L ) Q 0 ( 2 α L γ L ) ( α L δ L β L γ L ) ]    .
η tot = 4 2 + 2 ( α L δ L β L γ L ) 2 + 4 β L 2 δ L 2 + 4 α L 2 γ L 2    .
η tot = 1 ( α L 2 + β L 2 ) ( γ L 2 + δ L 2 ) .
a ( z ) = a 1 + ( a 2 a 1 ) z L ,
Ω ( t ) = ( 8 Δ ) 1 / 2 d a d z = ( 8 Δ ) 1 / 2 ( a 2 a 1 ) / L = constant  .
α ( z ) = Ψ 1  cos  [ Ψ t ( z ) + cos 1 Ψ ]  ,
β ( z ) = γ ( z ) = Ψ 1  sin [ Ψ t ( z ) ]  ,
δ ( z ) = Ψ 1  cos  [ Ψ t ( z ) cos 1 Ψ ]  ,
Ψ = [ 1 Ω 2 ] 1 / 2 ,
t ( z ) = ( 2 Δ ) 1 / 2 L ( a 2 a 1 )  ln [ a ( z ) a 1 ] .
[ A B C D ] = [ cos ( Q 0 z ) 1 Q 0 sin ( Q d z ) Q 0 sin ( Q 0 z ) cos ( Q 0 z ) ] .
η L = ( 1 Ω 2 ) ( 1 Ω 2 ) + Ω 2  sin 2 ( Ψ t )  ,
η tot = ( 1 Ω 2 ) 2 ( 1 Ω 2 ) 2 + 4 Ω 2  sin 4 ( Ψ t ) .
η L = ( Ω 2 1 ) ( Ω 2 1 ) + Ω 2 sinh 2 ( | Ψ | t ) ,
η tot = ( Ω 2 1 ) 2 ( Ω 2 1 ) 2 + 4 Ω 2 sinh 4 ( | Ψ | t ) .
d a d z a Δ β 2 π = Q a 2 π ,
u ( t ) = 0 t Ψ ( t ) d t = 0 t 1 Ω 2 ( t ) 1 / 2 d t .
α ( t ) = A ( t ) [ Ψ ( t ) Ψ ( 0 ) ] 1 / 2 × cos  [ u ( t ) + cos 1 Ψ ( 0 ) + cos 1 Ψ ( t ) 2 + μ ( t ) ]  ,
β ( t ) = ( t ) [ Ψ ( t ) Ψ ( 0 ) ] 1 / 2 × sin  [ u ( t ) cos 1 Ψ ( 0 ) cos 1 Ψ ( t ) 2 + ν ( t ) ]  ,
γ ( t ) = A ( t ) [ Ψ ( t ) Ψ ( 0 ) ] 1 / 2 ×  sin  [ u ( t ) + cos 1 Ψ ( 0 ) cos 1 Ψ ( t ) 2 + μ ( t ) ]  ,
δ ( t ) = B ( t ) [ Ψ ( t ) Ψ ( 0 ) ] 1 / 2 × cos  [ u ( t ) cos 1 Ψ ( 0 ) + cos 1 Ψ ( t ) 2 + ν ( t ) ]  ,
A ˙ A = + Ω ˙ 2 [ 1 Ω 2 ]  sin [ 2 u ( t ) + 2 μ ( t ) ]  ,
Ω ˙ = + μ ˙ 2 [ 1 Ω 2 ]  cos [ 2 u ( t ) + 2 μ ( t ) ]  ,
˙ = Ω ˙ 2 [ 1 Ω 2 ]  sin [ 2 u ( t ) + 2 ν ( t ) ]  ,
ν ˙ = Ω ˙ 2 [ 1 Ω 2 ]  cos [ 2 u ( t ) + 2 ν ( t ) ] .
α L = A L  cos [ u ( T ) + μ L ]  ,
β L = L  sin [ u ( T ) + ν L ]  ,
γ L = A L  sin [ u ( T ) + μ L ]  ,
δ L = L  cos  [ u ( T ) + ν L ] .
η L = 4 [ 2 + * * L 2 + L 2 ] .
d d u ln A = ( u ) sin ( 2 M )  ,
d M d u = 1 + d μ d u = 1 + d μ / d t d u / d t = 1 + ( u )  cos ( 2 M )  ,
( u ) = Ω ˙ 2 [ 1 Ω 2 ] 3 / 2 .
Ω ( t ) = 2 1 t [ 1 + ( 2 1 t ) 2 ] 1 / 2  ,
a ( t ) = a 1  exp  { 1 1 [ 1 + ( 2 1 t ) 2 ] 1 / 2 1 1 } .
Ω ( t ) = 2 2 ( T t ) { 1 + [ 2 2 ( T t ) ] 2 } 1 / 2  ,
a ( t ) = a 2  exp  { 1 2 1 2 [ 1 + [ 2 2 ( T t ) 2 ] 1 / 2 } .
I  ln  ( a I a 1 ) = 2  ln  ( a 2 a I )  ,
1 t 1 = 2 t 2 .
z ( t ) = 0 t a ( t ) [ 2 Δ ] 1 / 2 d t
L = ( a 2 a 1 ) [ 8 Δ ] 1 / 2 = 1.687  nm .
d d u [ ln  ( d M d u ) ] = d d u ln  A 2  ,
A 2 = constant d M / d u = constant 1 +  cos  ( 2 M ) .
2 = constant dN / du = constant 1  cos ( 2 N )  ,
A L 2 = A I 2 1 2  cos ( 2 M I ) 1 2  cos  ( 2 M L ) = 1 + 1 1 + 1  cos  ( 2 M I ) 1 2  cos  ( 2 M I ) 1 2  cos  ( 2 M L )  ,
1 1 1 + I  tan  ( M I ) =  tan  ( υ 1 ) ,
1 + 2 1 2  tan  ( M L ) =  tan  { υ 2 + tan 1 [ 1 + 2 1 + 2  tan  ( M I ) ] }  ,
υ 1 , 2 = [ 1 1 , 2 2 ] 1 / 2 u 1 , 2  ,
u 1 , 2 = [ 2 1 , 2 ] 1 / 2  ln  { [ 1 + ( 2 1 , 2 t 1 , 2 ) 2 ] 1 / 2 + 2 1 , 2 t 1 , 2 }
α L 2 = [ cos  ( υ 1 )  cos  ( υ 2 ) R 1 R 2  sin ( υ 1 )  sin ( υ 2 ) ] 2  ,
γ L 2 = [ 1 R 2  cos ( υ 1 )  sin ( υ 2 ) R 1  sin ( υ 1 )  cos ( υ 2 ) ] ] 2 .
R 1  , 2 = [ 1 + 1 , 2 1 1 , 2 ] 1 / 2  ,

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