Abstract

A method for the automated alignment of optical waveguides and fibers based on a multiobjective evolutionary algorithm is proposed. This algorithm reduces the number of parallel operations considerably compared to previous automation schemes. The automated alignment of a single-core input fiber with a channel waveguide and a single-core output fiber is completed using this system in less than 3min. The alignment of a single-core input fiber with a 1×8 splitter coupler and an eight-core output fiber array is completed in less than 10min. These results demonstrate the effectiveness of the proposed scheme for automated waveguide alignment, substantially outperforming previous automatic alignment methods.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. Y. Li and H. M. Niu, "The new development and trend of FTTH," Study Opt. Commun. 122, 8-10 (2004) (in Chinese).
  3. S. Hardy, "China weighs FTTH options," Lightwave 22, 8 (2005).
  4. A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).
  5. J. Gamet, "Field matching Y-branch for low loss power splitter," Opt. Commun. 248, 423-430 (2005).
    [CrossRef]
  6. K. Murphy and J. Edmond, "Fiber-waveguide self alignment coupler," U.S. patent 4,639,074 (27 January 1987).
  7. R. Germann, H. Horak, and V. David, "Self-aligned optical waveguide to optical fiber connection system," U.S. patent 6,819,841 (16 November 2004).
  8. B. X. Chen, C. H. Long, and H. J. Sha, "Automatic alignment between waveguide and fiber based on centroid arithmetic," Chinese patent ZL03129249.6 (2005) (in Chinese).
  9. C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).
  10. M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
    [CrossRef]
  11. H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).
  12. R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).
  13. Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).
  14. W. P. Chen and L. S. Kang, "Multi-parent-crossover based evolutionary multi-objective optimization," Comput. Eng. Appl. 39, 79-82 (2003) (in Chinese).
  15. C. Fonseca and P. J. Fleming, "Multi-objective optimization and multiple constraint handling with evolutionary algorithms part I: a unified formulation," IEEE Trans. Syst. Man Cybern. 28, 26-37 (1998).
    [CrossRef]
  16. K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
    [CrossRef]

2006

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

2005

S. Hardy, "China weighs FTTH options," Lightwave 22, 8 (2005).

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

J. Gamet, "Field matching Y-branch for low loss power splitter," Opt. Commun. 248, 423-430 (2005).
[CrossRef]

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

2004

Y. Li and H. M. Niu, "The new development and trend of FTTH," Study Opt. Commun. 122, 8-10 (2004) (in Chinese).

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
[CrossRef]

2003

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

W. P. Chen and L. S. Kang, "Multi-parent-crossover based evolutionary multi-objective optimization," Comput. Eng. Appl. 39, 79-82 (2003) (in Chinese).

1998

C. Fonseca and P. J. Fleming, "Multi-objective optimization and multiple constraint handling with evolutionary algorithms part I: a unified formulation," IEEE Trans. Syst. Man Cybern. 28, 26-37 (1998).
[CrossRef]

1975

Agrawal, S.

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
[CrossRef]

Baby, A.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Chen, B. X.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

B. X. Chen, C. H. Long, and H. J. Sha, "Automatic alignment between waveguide and fiber based on centroid arithmetic," Chinese patent ZL03129249.6 (2005) (in Chinese).

Chen, L.

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

Chen, W. P.

W. P. Chen and L. S. Kang, "Multi-parent-crossover based evolutionary multi-objective optimization," Comput. Eng. Appl. 39, 79-82 (2003) (in Chinese).

Chen, Y. P.

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

Cheng, L.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

David, V.

R. Germann, H. Horak, and V. David, "Self-aligned optical waveguide to optical fiber connection system," U.S. patent 6,819,841 (16 November 2004).

Deb, K.

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
[CrossRef]

Dhanavantri, C.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Edmond, J.

K. Murphy and J. Edmond, "Fiber-waveguide self alignment coupler," U.S. patent 4,639,074 (27 January 1987).

Fleming, P. J.

C. Fonseca and P. J. Fleming, "Multi-objective optimization and multiple constraint handling with evolutionary algorithms part I: a unified formulation," IEEE Trans. Syst. Man Cybern. 28, 26-37 (1998).
[CrossRef]

Fonseca, C.

C. Fonseca and P. J. Fleming, "Multi-objective optimization and multiple constraint handling with evolutionary algorithms part I: a unified formulation," IEEE Trans. Syst. Man Cybern. 28, 26-37 (1998).
[CrossRef]

Fu, P. H.

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

Gamet, J.

J. Gamet, "Field matching Y-branch for low loss power splitter," Opt. Commun. 248, 423-430 (2005).
[CrossRef]

Germann, R.

R. Germann, H. Horak, and V. David, "Self-aligned optical waveguide to optical fiber connection system," U.S. patent 6,819,841 (16 November 2004).

Guo, Y.

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

Guttmann, J.

Hardy, S.

S. Hardy, "China weighs FTTH options," Lightwave 22, 8 (2005).

Higuchi, T.

M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
[CrossRef]

Horak, H.

R. Germann, H. Horak, and V. David, "Self-aligned optical waveguide to optical fiber connection system," U.S. patent 6,819,841 (16 November 2004).

Iso, M.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

Johri, S.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Kang, L. S.

W. P. Chen and L. S. Kang, "Multi-parent-crossover based evolutionary multi-objective optimization," Comput. Eng. Appl. 39, 79-82 (2003) (in Chinese).

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

Krumpholz, O.

Kumar, P.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Li, Y.

Y. Li and H. M. Niu, "The new development and trend of FTTH," Study Opt. Commun. 122, 8-10 (2004) (in Chinese).

Long, C. H.

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

B. X. Chen, C. H. Long, and H. J. Sha, "Automatic alignment between waveguide and fiber based on centroid arithmetic," Chinese patent ZL03129249.6 (2005) (in Chinese).

Meyarivan, T.

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
[CrossRef]

Murakawa, M.

M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
[CrossRef]

Murphy, K.

K. Murphy and J. Edmond, "Fiber-waveguide self alignment coupler," U.S. patent 4,639,074 (27 January 1987).

Niu, H. M.

Y. Li and H. M. Niu, "The new development and trend of FTTH," Study Opt. Commun. 122, 8-10 (2004) (in Chinese).

Noda, Y.

M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
[CrossRef]

Pachauri, J. P.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Pfeiffer, E.

Pratap, A.

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
[CrossRef]

Sha, H. J.

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

B. X. Chen, C. H. Long, and H. J. Sha, "Automatic alignment between waveguide and fiber based on centroid arithmetic," Chinese patent ZL03129249.6 (2005) (in Chinese).

Singh, B. R.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

Sui, G. R.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

Sun, R. X.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

Yan, Z. Y.

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

Yuan, Y. F.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

Zhang, X. W.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

Zhao, D. X.

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

Acta Opt. Sin.

C. H. Long, B. X. Chen, H. J. Sha, D. X. Zhao, Y. Guo, Y. F. Yuan, and M. Iso, "Study on fiber-waveguide automatic alignment system," Acta Opt. Sin. 24, 442-447 (2004) (in Chinese).

Appl. Opt.

Chin. J. Sci. Instrum.

H. J. Sha, B. X. Chen, L. Chen, Y. F. Yuan, and M. Iso, "The study of fiber-waveguide automatic alignment system based on genetic algorithm," Chin. J. Sci. Instrum. 26, 1109-1113 (2005) (in Chinese).

Comput. Eng. Appl.

W. P. Chen and L. S. Kang, "Multi-parent-crossover based evolutionary multi-objective optimization," Comput. Eng. Appl. 39, 79-82 (2003) (in Chinese).

IEEE Trans. Syst. Man Cybern.

C. Fonseca and P. J. Fleming, "Multi-objective optimization and multiple constraint handling with evolutionary algorithms part I: a unified formulation," IEEE Trans. Syst. Man Cybern. 28, 26-37 (1998).
[CrossRef]

Ind. J. Eng. Mater. Sci.

A. Baby, C. Dhanavantri, J. P. Pachauri, S. Johri, P. Kumar, and B. R. Singh, "Silica-on-silicon based 1 × N optical splitter: design, fabrication and characterization," Ind. J. Eng. Mater. Sci. 12, 12-16 (2005).

J. WuHan Univ.

Z. Y. Yan, L. S. Kang, Y. P. Chen, and P. H. Fu, "A new multi-objective evolutionary algorithm: steady elimination evolutionary algorithm," J. WuHan Univ. (Nat. Sci. Ed.) 49, 33-38 (2003) (in Chinese).

Lightwave

S. Hardy, "China weighs FTTH options," Lightwave 22, 8 (2005).

Opt. Commun.

J. Gamet, "Field matching Y-branch for low loss power splitter," Opt. Commun. 248, 423-430 (2005).
[CrossRef]

Opt. Tech.

R. X. Sun, B. X. Chen, G. R. Sui, X. W. Zhang, L. Cheng, Y. F. Yuan, and M. Iso, "Automatic alignment between waveguide array and fiber array based on genetic algorithm," Opt. Tech. 32, 533-536 (2006) (in Chinese).

Study Opt. Commun.

Y. Li and H. M. Niu, "The new development and trend of FTTH," Study Opt. Commun. 122, 8-10 (2004) (in Chinese).

Syst. Comput. Japan

M. Murakawa, Y. Noda, and T. Higuchi, "An automatic fiber alignment system using genetic algorithms," Syst. Comput. Japan 35, 80-90 (2004).
[CrossRef]

Other

K. Murphy and J. Edmond, "Fiber-waveguide self alignment coupler," U.S. patent 4,639,074 (27 January 1987).

R. Germann, H. Horak, and V. David, "Self-aligned optical waveguide to optical fiber connection system," U.S. patent 6,819,841 (16 November 2004).

B. X. Chen, C. H. Long, and H. J. Sha, "Automatic alignment between waveguide and fiber based on centroid arithmetic," Chinese patent ZL03129249.6 (2005) (in Chinese).

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, "A fast elitist nondominated sorting genetic algorithm for multi-objective optimization: NSGA-II," in Parallel Problem Solving from Nature-PPSN VI (Springer, 2000), pp. 849-858.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Analytic model of the end-coupling between a single-mode fiber and a single-mode waveguide.

Fig. 2
Fig. 2

Alignment between a 1 × 8 waveguide splitter and a fiber array.

Fig. 3
Fig. 3

(Color online)Process characteristic of mimic alignment based on a multiobjective evolutionary algorithm.

Fig. 4
Fig. 4

Insertion loss of eight channels.

Fig. 5
Fig. 5

(Color online) Construction of automatic alignment system.

Tables (5)

Tables Icon

Table 1 Simulation Results of Two-Channel Alignment between an Idealized Fiber Array and a Waveguide Array

Tables Icon

Table 2 Actual Position of Eight Core Centers in the Fiber Array

Tables Icon

Table 3 Mimic Results of Two-Channel Alignment between the Actual Fiber Array and the Waveguide Splitter

Tables Icon

Table 4 Experimental Results of the Fiber–Waveguide–Fiber System at 1542.4 nm Wavelength

Tables Icon

Table 5 Experimental Results of the Single-Core Fiber Array—1 × 8 Splitter Coupler—Eight-Core Fiber Array System

Equations (148)

Equations on this page are rendered with MathJax. Learn more.

3 min
1 × 8
10 min
1 × N
1 × N
1 × 8
10   dB
9   dB
250   μm
0 .5   μm
0 .1   μm
0 .1   μm
2 / M
50   dB
( θ )
Δ x
Δ y
W x o
W y o
W f o
( η )
η = η x η y ,
η x
η y
η x = 2 exp [ k 2 ( W f o 2 + W x o 2 ) θ 2 2 η x o ( Δ z ) 2 η x o ( Δ z ) ( Δ x 2 W f o 2 + ( Δ z θ Δ x ) 2 W x o 2 ) ] η x o ( Δ z ) ,
η y = 2 exp [ 2 Δ y 2 η y o ( Δ z ) ( 1 W f o 2 + 1 W y o 2 ) ] η y o ( Δ z ) .
η x o ( Δ z ) = ( W x o W f o + W f o W x o ) 2 + ( 2 Δ z k W f o W x o ) 2 ,
η y o ( Δ z ) = ( W y o W f o + W f o W y o ) 2 + ( 2 Δ z k W f o W y o ) 2 ,
Δ x
Δ y
Δ z
Δ x
Δ y
Δ z
η max = 4 ( W f o W x o + W x o W f o ) ( W f o W y o + W y o W f o ) .
100 %
( < 0.1   μm )
( θ x , θ y , z )
θ z
θ z
M + 1
50   μm
4   μm
50 50   μm
0 4   μm
250   μm
1250   μm
θ z
5 °
+ 5 °
θ x
θ y
z max h sin θ x z max h , z max w sin θ y z max w ,
z max
U i
f 1 ( U i ) = [ P 1 ( U i ) + P 2 ( U i ) ] ,
f 2 ( U i ) = | P 1 ( U i ) P 2 ( U i ) | ,
P 1
P 2
U i
P 1
P 2
P 1
P 2
f 1
P 1
P 2
f 2
U i
U i
U j i
f 1 ( U i ) > f i ( U j i )   and   f 2 ( U i ) f 2 ( U j i ) ,
U i
U i
f 1 ( U m ) < f 1 ( U m + 1 ) m = 1 , 2 ,   .   .   .   ,   k 1 , f 2 ( U n ) < f 2 ( U n + 1 ) n = 1 , 2 ,   .   .   .   ,   k 1.
d 1 ( U m ) = { at   m = 1   or   k f 1 ( U m + 1 ) f 1 ( U m 1 ) f 1 ( U k ) f 1 ( U 1 ) at   m 1   or   k .
d 2 ( U n ) = { at   n = 1   or   k f 2 ( U n + 1 ) f 2 ( U n 1 ) f 2 ( U k ) f 2 ( U 1 ) at   n 1   or   k .
d 1 ( U i )
d 2 ( U i )
U i
d ( U i ) = d 1 ( U i ) + d 2 ( U i ) .
U 0 = i = 1 m a i U i .
U 0
i = 1 m a i = 1 , 0.5 a i 1.5.
M + 1
U 0
U 0
M + 1
1 × 8
9.031   dB
100%
5 .63   μm
1550   nm
0 .1%
α = 1 ( W x o / W y o )
Δ = 0.3 %
0.4 %
W y o
5.63 μ m
W x o
5.61   μm
0   dBm
250   μm
1250   μm
1550   nm
98% / m m
9 .042   dB
0 .011   dB
0   dB
9 .1   dB
0 .35   μm
250   μm
9 .050   dB
9 .085   dB
0 .008   dB
0 .043   dB
9.072   dB
0 .041   dB
0 .054   dB
0 .05   μm
1 / 20
θ x
θ y
θ z
8 × 8 μm 2
0.3 %
± 50   μm
4   μm
3   min
( 50   μm )
1542.4   nm
0 .1136   dB
0 .13   dB
0 .02   dB
1 × 8
250   m
1542 .4   nm
10   min
10 .7   dB
0 .77   dB
3   min
0 .1136   dB
0 .13   dB
0 .02   dB
1 × 8
10 .7   dB
0 .77   dB
1 × 8

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