Abstract

We present a multilevel global optimization strategy for synthesizing planar multilayered dielectric structures. A low discrepancy sequence of sample points with uniform variable space coverage allows a global-level search while systematic refinement using gradient-based techniques identifies optima at the local level. Since efficient local optimization is important for this method, a fast calculation approach based on mode matching is presented; this also facilitates the compact derivation of analytical gradients. The approach is compared with genetic and simulated annealing algorithms through an antireflection coating design. The method proves to be competitive in terms of its performance, nonadaptive algorithm, and ability to track local solutions.

© 2006 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th (revised) ed. (Cambridge U. Press, 1999).
  2. H. A. McLeod, Thin Film Optical Filters, 3rd ed. (Adam Hilger, 2001).
    [CrossRef]
  3. J. A. Dobrowolski and R. A. Kemp, "Refinement of optical multilayer systems with different optimization procedures," Appl. Opt. 29, 2876-2892 (1990).
    [CrossRef] [PubMed]
  4. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
    [CrossRef]
  5. S. Martin, J. Rivory, and M. Schoenauer, "Synthesis of optical multilayer systems using genetic algorithms," Appl. Opt. 34, 2247-2254 (1995).
    [CrossRef] [PubMed]
  6. V. Yakovelev and G. Tempea, "Optimization of chirped mirrors," Appl. Opt. 41, 6514-6520 (2002).
    [CrossRef]
  7. C.-L. Lee and Y. Lai, "Optical narrowband dispersionless fiber Bragg grating filter with short grating length and smooth dispersion profile," Opt. Commun. 235, 99-106 (2004).
    [CrossRef]
  8. T. Boudet, P. Chaton, L. Herault, G. Gonon, L. Jouanet, and P. Keller, "Thin-film designs by simulated annealing," Appl. Opt. 35, 6219-6222 (1996).
    [CrossRef] [PubMed]
  9. L. Liberti and S. Kucherenko, "Comparison of deterministic and stochastic approaches to global optimization," Int. Trans. Oper. Res. 12, 263-281 (2005).
    [CrossRef]
  10. J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).
  11. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
    [CrossRef] [PubMed]
  12. A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part i, clustering methods," Math. Program. 39, 27-56 (1987).
    [CrossRef]
  13. A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part ii, multilevel methods," Math. Program. 39, 57-78 (1987).
    [CrossRef]
  14. S. Kucherenko and Y. Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Comput. Optim. Appl. 30, 297-318 (2005).
    [CrossRef]
  15. I. M. Sobol, "On the distribution of points in a cube and the approximate evaluation of integrals," Comput. Math. Phys. 7, 86-112 (1992).
    [CrossRef]
  16. P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
    [CrossRef]
  17. P. J. B. Clarricoats and K. R. Slinn, "Numerical method for the solution of waveguide discontinuity problems," Electron. Lett. 2, 226-239 (1966).
    [CrossRef]
  18. P. J. B. Clarricoats and K. R. Slinn, "Numerical solution of waveguide discontinuity problems," Proc. IEEE 114, 878-886 (1967).
  19. A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE Trans. MTT 9, 508-516 (1967).
    [CrossRef]
  20. K.-O. Peng and M. R. de la Fonteijne, "Derivatives of transmittance and reflectance for an absorbing multilayer stack," Appl. Opt. 24, 501-503 (1985).
    [CrossRef] [PubMed]
  21. T. I. Oh, "Broadband AR coatings on germanium substrates using ion assisted deposition," Appl. Opt. 27, 4255-4259 (1988).
    [CrossRef] [PubMed]
  22. J. H. Holton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals," Numer. Math. 2, 84-90 (1960).
    [CrossRef]
  23. H. Niederreiter, "Point sets and sequences with small discrepancy," Monatsh. Math. 104, 273-337 (1987).
    [CrossRef]
  24. R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J. 6, 163-168 (1963).
  25. T. F. Coleman and Y. Li, "An interior, trust region approach for non-linear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
    [CrossRef]
  26. J. A. Dobrowolski, A. V. Tikhonaravov, M. K. Trubetskov, B. T. Sullivan, and P. G. Verly, "Refinement of optical multilayer systems with different optimization procedures," Appl. Opt. 35, 644-658 (1996).
    [CrossRef] [PubMed]
  27. A. V. Tikhonaravov and J. A. Dobrowolski, "A new, quasi-optimal synthesis method for antireflection coatings," Appl. Opt. 32, 4265-4275 (1993).
    [CrossRef]

2005 (2)

L. Liberti and S. Kucherenko, "Comparison of deterministic and stochastic approaches to global optimization," Int. Trans. Oper. Res. 12, 263-281 (2005).
[CrossRef]

S. Kucherenko and Y. Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Comput. Optim. Appl. 30, 297-318 (2005).
[CrossRef]

2004 (1)

C.-L. Lee and Y. Lai, "Optical narrowband dispersionless fiber Bragg grating filter with short grating length and smooth dispersion profile," Opt. Commun. 235, 99-106 (2004).
[CrossRef]

2002 (1)

1996 (3)

1995 (1)

1994 (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
[CrossRef]

1993 (1)

1992 (2)

I. M. Sobol, "On the distribution of points in a cube and the approximate evaluation of integrals," Comput. Math. Phys. 7, 86-112 (1992).
[CrossRef]

P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
[CrossRef]

1990 (1)

1988 (1)

1987 (3)

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part i, clustering methods," Math. Program. 39, 27-56 (1987).
[CrossRef]

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part ii, multilevel methods," Math. Program. 39, 57-78 (1987).
[CrossRef]

H. Niederreiter, "Point sets and sequences with small discrepancy," Monatsh. Math. 104, 273-337 (1987).
[CrossRef]

1985 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

1967 (2)

P. J. B. Clarricoats and K. R. Slinn, "Numerical solution of waveguide discontinuity problems," Proc. IEEE 114, 878-886 (1967).

A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE Trans. MTT 9, 508-516 (1967).
[CrossRef]

1966 (1)

P. J. B. Clarricoats and K. R. Slinn, "Numerical method for the solution of waveguide discontinuity problems," Electron. Lett. 2, 226-239 (1966).
[CrossRef]

1963 (1)

R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J. 6, 163-168 (1963).

1960 (1)

J. H. Holton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals," Numer. Math. 2, 84-90 (1960).
[CrossRef]

Anemogiannis, E.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th (revised) ed. (Cambridge U. Press, 1999).

Boudet, T.

Bratley, P.

P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
[CrossRef]

Chaton, P.

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. R. Slinn, "Numerical solution of waveguide discontinuity problems," Proc. IEEE 114, 878-886 (1967).

P. J. B. Clarricoats and K. R. Slinn, "Numerical method for the solution of waveguide discontinuity problems," Electron. Lett. 2, 226-239 (1966).
[CrossRef]

Coleman, T. F.

T. F. Coleman and Y. Li, "An interior, trust region approach for non-linear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

de la Fonteijne, M. R.

Dobrowolski, J. A.

Fletcher, R.

R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J. 6, 163-168 (1963).

Fox, B. L.

P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
[CrossRef]

Gaylord, T. K.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Glytsis, E. N.

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
[CrossRef]

Gonon, G.

Herault, L.

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).

Holton, J. H.

J. H. Holton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals," Numer. Math. 2, 84-90 (1960).
[CrossRef]

Jouanet, L.

Kan, A. H. G. Rinnooy

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part i, clustering methods," Math. Program. 39, 27-56 (1987).
[CrossRef]

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part ii, multilevel methods," Math. Program. 39, 57-78 (1987).
[CrossRef]

Keller, P.

Kemp, R. A.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Kucherenko, S.

L. Liberti and S. Kucherenko, "Comparison of deterministic and stochastic approaches to global optimization," Int. Trans. Oper. Res. 12, 263-281 (2005).
[CrossRef]

S. Kucherenko and Y. Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Comput. Optim. Appl. 30, 297-318 (2005).
[CrossRef]

Lai, Y.

C.-L. Lee and Y. Lai, "Optical narrowband dispersionless fiber Bragg grating filter with short grating length and smooth dispersion profile," Opt. Commun. 235, 99-106 (2004).
[CrossRef]

Lee, C.-L.

C.-L. Lee and Y. Lai, "Optical narrowband dispersionless fiber Bragg grating filter with short grating length and smooth dispersion profile," Opt. Commun. 235, 99-106 (2004).
[CrossRef]

Li, Y.

T. F. Coleman and Y. Li, "An interior, trust region approach for non-linear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

Liberti, L.

L. Liberti and S. Kucherenko, "Comparison of deterministic and stochastic approaches to global optimization," Int. Trans. Oper. Res. 12, 263-281 (2005).
[CrossRef]

Martin, S.

McLeod, H. A.

H. A. McLeod, Thin Film Optical Filters, 3rd ed. (Adam Hilger, 2001).
[CrossRef]

Neiderreiter, H.

P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
[CrossRef]

Niederreiter, H.

H. Niederreiter, "Point sets and sequences with small discrepancy," Monatsh. Math. 104, 273-337 (1987).
[CrossRef]

Oh, T. I.

Peng, K.-O.

Powell, M. J. D.

R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J. 6, 163-168 (1963).

Rivory, J.

Schoenauer, M.

Slinn, K. R.

P. J. B. Clarricoats and K. R. Slinn, "Numerical solution of waveguide discontinuity problems," Proc. IEEE 114, 878-886 (1967).

P. J. B. Clarricoats and K. R. Slinn, "Numerical method for the solution of waveguide discontinuity problems," Electron. Lett. 2, 226-239 (1966).
[CrossRef]

Sobol, I. M.

I. M. Sobol, "On the distribution of points in a cube and the approximate evaluation of integrals," Comput. Math. Phys. 7, 86-112 (1992).
[CrossRef]

Sullivan, B. T.

Sytsko, Y.

S. Kucherenko and Y. Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Comput. Optim. Appl. 30, 297-318 (2005).
[CrossRef]

Tempea, G.

Tikhonaravov, A. V.

Timmer, G. T.

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part i, clustering methods," Math. Program. 39, 27-56 (1987).
[CrossRef]

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part ii, multilevel methods," Math. Program. 39, 57-78 (1987).
[CrossRef]

Trubetskov, M. K.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Verly, P. G.

Wexler, A.

A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE Trans. MTT 9, 508-516 (1967).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th (revised) ed. (Cambridge U. Press, 1999).

Yakovelev, V.

ACM Trans. Model. Comput. Simul. (1)

P. Bratley, B. L. Fox, and H. Neiderreiter, "Implementation and tests of low discrepancy sequences," ACM Trans. Model. Comput. Simul. 2, 195-213 (1992).
[CrossRef]

Appl. Opt. (8)

Comput. J. (1)

R. Fletcher and M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J. 6, 163-168 (1963).

Comput. Math. Phys. (1)

I. M. Sobol, "On the distribution of points in a cube and the approximate evaluation of integrals," Comput. Math. Phys. 7, 86-112 (1992).
[CrossRef]

Comput. Optim. Appl. (1)

S. Kucherenko and Y. Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Comput. Optim. Appl. 30, 297-318 (2005).
[CrossRef]

Electron. Lett. (1)

P. J. B. Clarricoats and K. R. Slinn, "Numerical method for the solution of waveguide discontinuity problems," Electron. Lett. 2, 226-239 (1966).
[CrossRef]

IEEE Trans. MTT (1)

A. Wexler, "Solution of waveguide discontinuities by modal analysis," IEEE Trans. MTT 9, 508-516 (1967).
[CrossRef]

Int. Trans. Oper. Res. (1)

L. Liberti and S. Kucherenko, "Comparison of deterministic and stochastic approaches to global optimization," Int. Trans. Oper. Res. 12, 263-281 (2005).
[CrossRef]

J. Lightwave Technol. (1)

E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, "Optimization of multilayer integrated optics waveguides," J. Lightwave Technol. 12, 512-517 (1994).
[CrossRef]

Math. Program. (2)

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part i, clustering methods," Math. Program. 39, 27-56 (1987).
[CrossRef]

A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimisation methods, part ii, multilevel methods," Math. Program. 39, 57-78 (1987).
[CrossRef]

Monatsh. Math. (1)

H. Niederreiter, "Point sets and sequences with small discrepancy," Monatsh. Math. 104, 273-337 (1987).
[CrossRef]

Numer. Math. (1)

J. H. Holton, "On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals," Numer. Math. 2, 84-90 (1960).
[CrossRef]

Opt. Commun. (1)

C.-L. Lee and Y. Lai, "Optical narrowband dispersionless fiber Bragg grating filter with short grating length and smooth dispersion profile," Opt. Commun. 235, 99-106 (2004).
[CrossRef]

Proc. IEEE (1)

P. J. B. Clarricoats and K. R. Slinn, "Numerical solution of waveguide discontinuity problems," Proc. IEEE 114, 878-886 (1967).

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

SIAM J. Optim. (1)

T. F. Coleman and Y. Li, "An interior, trust region approach for non-linear minimization subject to bounds," SIAM J. Optim. 6, 418-445 (1996).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 6th (revised) ed. (Cambridge U. Press, 1999).

H. A. McLeod, Thin Film Optical Filters, 3rd ed. (Adam Hilger, 2001).
[CrossRef]

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a multilayer structure showing (a) the layers with labels of interface positions, z i ; layer thickness, d i ; and refractive indices, n i . (b), (c) Eigenvector solutions of the wave equation used in the mode-matching model for the TE (s) and TM (p) polarizations, respectively.

Fig. 2
Fig. 2

Distributions of starting points in a 50-dimensional unit hypercube generated using (a)–(d) Sobol's low-discrepancy sequence and (e)–(h) a pseudorandom number generator. Uniformity of the generation methods is examined for 64 and 1024 start points. Here the projections (a), (c), (e), and (g) show dimensions 1 and 2 and (b), (d), (f), and (h) show dimensions 49 and 50.

Fig. 3
Fig. 3

Plots showing the frequency of convergence to a local solution for (a) 8-period DBR (N = 15 variables), (b) 24-period DBR (N = 47 variables), and (c) 48-period DBR (N = 95 variables). The plots compare the distributions for both the SQP (gray bars) and trust-region (black bars) search methods. Although both methods perform similarly for lower dimensions, accuracy and resolution of the various local solutions become a problem at high dimensions for the SQP method.

Fig. 4
Fig. 4

Results of global optimization for a range of DBRs with 1000 Sobol starting points. (a) Proportion of Sobol starting points that converged to the global optimum. (b) Number of local solutions identified. (Solutions where solver reached the iteration limit have been omitted.) (c) Total number of function evaluations during the global optimization run. TR, trust region.

Fig. 5
Fig. 5

Reflectivity spectra for (a) 17-layer and (b) 20-layer antireflection coatings optimized using the multilevel algorithm described in Subsection 4.C, a genetic algorithm (Ref. 5), and simulated annealing (Ref. 8). (c) Positions of all the solutions for 17-layer (dots) and 20-layer (crosses) antireflection coatings plotted as a function of the merit function, F , and optical thickness. The best solutions are close to the dashed curve, which is the theoretical optimal locus for the antireflection coating considered here (Ref. 26).

Tables (2)

Tables Icon

Table 1 Antireflection Coating Designs Found by Multilevel Optimization

Tables Icon

Table 2 Optimum Antireflection Coating Designs by Multilevel Method

Equations (36)

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min x F ( x , y ) ,
x - x U 0 ,
x L - x 0 ,
k i × k i × E i = k 0 2 ε i E i .
( k 0 2 ε i k y 2 β i     2 k x k y k x β i k x k y k 0 2 ε i k x 2 β i     2 k y β i k x β i k y β i k 0 2 ε i k x 2 k y 2 ) × ( E x E y E z ) = 0.
E i = N i { a s i e s i + a ¯ s i e ¯ s i + a p i e p i + a ¯ p i e ¯ p i }  exp ( i k i r ) ,
H i = N i { a s i h s i + a ¯ s i h ¯ si + a p i h p i + a ¯ p i h ¯ pi } exp ( i k i r ) ,
β i = ± k 0 2 n i 2 k x 2 ,
e s i = { 0 , 1 , 0 } ,
e p i = { β i n i k 0 , 0 , k x n i k 0 } .
( a s i + a ¯ s i ) s i + ( a p i + a ¯ p i ) p i = ( a s j + a ¯ s j ) s j + ( a p j + a ¯ p j ) p j ,
( a s i a ¯ s i ) s i + ( a p i a ¯ p i ) p i = ( a s j a ¯ s j ) s j + ( a p j a ¯ p j ) p j .
( a s j + a ¯ s j ) C j , j ( s s ) + ( a p j + a ¯ p j ) C j , j ( p s ) = ( a s i + a ¯ s i ) C j , j ( s s ) + ( a p i + a ¯ p i ) C i , j ( p s ) ,
( a s j + a ¯ s j ) C j , j ( s p ) + ( a p j + a ¯ p j ) C j , j ( p p ) = ( a s i + a ¯ s i ) C i , j ( s p ) + ( a p i + a ¯ p i ) C i , j ( p p ) ,
( a s j a ¯ s j ) C i , j ( s s ) + ( a p j a ¯ p j ) C i , j ( s p ) = ( a s i a ¯ s i ) C i , i ( s s ) + ( a p i a ¯ p i ) C i , i ( s p ) ,
( a s j a ¯ s j ) C i , j ( p s ) + ( a p j a ¯ p j ) C i , j ( p p ) = ( a s i a ¯ s i ) C i , i ( p s ) + ( a p i a ¯ p i ) C i , i ( p p ) ,
C i , j ( μ ν ) = μ i × ν j z ^ d x d y = N i N j z 0 k 0 e μ i × k ν j × e ν j z ^ δ ( k μ i x ^ k ν j x ^ ) ,
N i = z 0 k 0 e μ i × k μ i × e μ i · z ^ = z 0 k 0 β i .
C i , j ( s s ) = k z j k z i     C i , j ( s s ) = n j n i k z i k z j .
( 1 1 0 0 C i , j ( s s ) - C i , j ( s s ) 0 0 0 0 1 1 0 0 C i , j ( p p ) - C i , j ( p p ) ) ( a s j a ¯ s j a p j a ¯ p j ) = ( C i , j ( s s ) C i , j ( s s ) 0 0 1 1 0 0 0 0 C i , j ( p p ) - C i , j ( p p ) 0 0 1 1 ) ( a s i a ¯ s i a p i a ¯ p i ) .
( a j a ¯ j ) = ( X j , i + X j , i - X j , i - X j , i + ) ( a i a ¯ i ) = T j i ( a i a ¯ i ) ,
X j , i ( μ ) ± = 1 2 ( C i , j ( μ ) ± 1 C i , j ( μ ) ) .
( a j ( z j ) a ¯ j ( z j ) ) = ( exp [ i β j ( z j z i ) ] 0 0 exp [ i β j ( z j z i ) ] ) ( a j ( z i ) a ¯ j ( z i ) ) = P j ( a j ( z i ) a ¯ j ( z i ) ) .
( a i + 1 ( z i + 1 ) a ¯ i + 1 ( z i + 1 ) ) = P i + 1 T i +1, i ( a i ( z i ) a ¯ i ( z i ) ) = M i +1, i ( a i ( z i ) a ¯ i ( z i ) ) .
ζ i ( a N a ¯ N ) = M N , 1 ζ i ( a 1 a ¯ 1 ) + M N , 1 ζ i ( a 1 a ¯ 1 ) .
M N , 1 d i = M N , i M i , i 1 d i M i 1 , 1 = M N , i d M d i M i , 1 = M N , i [ i k z i 0 0 i k z i ] M i , 1 .
M N , 1 n i = M N , i + 1 P i + 1 n i ( T i + 1 , i P i T i , i 1 ) M i 1 , 1 .
T j , i ( μ ) n i = 1 2 ( 0 1 1 0 ) T j , i ( μ ) 1 C i , j ( μ ) C i , j ( μ ) n i ,
T i , j ( μ ) n i = 1 2 ( 0 1 1 0 ) T i , j ( μ ) 1 C j , i ( μ ) C j , i ( μ ) n i ,
P i n i = i n i d i k 0 2 β i ( 1 0 0 1 ) P i .
      1 C i , j ( s ) C i , j ( s ) n i = n i k 0 2 β i 2 = 1 C j , i ( s ) C j , i ( s ) n i ,
1 C i , j ( p ) C i , j ( p ) n i = β i 2 k x 2 β i 2 n i = 1 C j , i ( p ) C j , i ( p ) n i .
M N , 1 n i = M N , i d M n i M i , 1 ,
dM n i ( s ) = n i k 0 2 β i 2 [ i β i d i 1 2 { exp ( 2 i β i d i ) 1 } 1 2 { exp ( 2 i β i d i ) 1 } i β i d i ] dM n i ( p ) = n i k 0 2 β i 2 [ i β i d i 1 2 β i 2 k x 2 k 0 2 n i 2 { exp ( 2 i β i d i ) 1 } 1 2 β i 2 k x 2 k 0 2 n i 2 { exp ( 2 i β i d i ) 1 } i β i d i ] .
d R d ζ i = d [ a i ( z i ) a i ( z i ) * ] d ζ i = 2 { a i ( z i ) } dℜ { a i ( z i ) } d ζ i + 2 { a i ( z i ) } d { a i ( z i ) } d ζ i .
( d , λ i , R i ) = [ 1 N λ i = 1 N λ R i ( λ i ) 2 ] 1 / 2 .

Metrics