Abstract

The solution of the Helmholtz equation requires the application of an exponentiated square root operator to an initial field. This operation is greatly facilitated by the introduction of a representation in which the above-mentioned operator is diagonal. The Lanczos method permits this diagonalization to be performed in a low-dimensional space and the propagation to be carried out to arbitrary order. An iteration scheme to be carried out in conjunction with the Lanczos method is also described. This scheme permits, in principle, the bound mode of a monomode waveguide to be calculated to machine accuracy. Results for some well-known test examples of rib waveguides are presented.

© 1992 Optical Society of America

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References

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  1. R. P. Ratowsky, J. A. Fleck, “Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction,” Opt. Lett. 16, 787–789 (1991).
    [CrossRef] [PubMed]
  2. T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,”J. Chem. Phys. 85, 5870–5876 (1986).
    [CrossRef]
  3. C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
    [CrossRef]
  4. See, for example, W. F. Ames, D. Lee, “Current development in numerical treatment of ocean acoustic propagation,” Appl. Num. Math. 3, 25–47 (1987).
    [CrossRef]
  5. M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).
  6. B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,”IEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984); “Vector H finite element solutions of GaAs/GaAIAs rib waveguides,” Proc. Inst. Electr. Eng. Part J 132, 349–353 (1985).
    [CrossRef]
  7. M. S. Stern, “Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles,”IEE Proc. Part J 135, 56–63 (1988); M. S. Stern, “Semivectorial polarised Hfield solutions for dielectric waveguides with arbitrary index profiles,” Proc. Inst. Electr. Eng. Part J. 135, 333–338 (1988).
  8. N. Dagli, C. Fonstad, “Universal design curves for rib waveguides,” IEEE J. Lightwave Technol. 6, 1136–1145 (1988).
    [CrossRef]
  9. D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
    [CrossRef]
  10. M. D. Feit, J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. Am. A 7, 73–79 (1990).
    [CrossRef]
  11. W. P. Huang, H. A. Haus, “A simple variational approach to optical rib waveguides,” IEEE J. Lightwave Technol. 9, 56–61 (1991).
    [CrossRef]
  12. M. D. Feit, J. A. Fleck, “Computation of mode properties in optical fiber waveguides by the propagating beam method,” Appl. Opt. 19, 1154–1164 (1980).
    [CrossRef] [PubMed]
  13. The method of lines also works by matrix diagonalization, although the matrices are not so large as those envisaged here. See J. Gerdes, R. Pregla, “Beam propagation algorithm based on the method of lines,” J. Opt. Soc. Am. B 8, 389–395 (1991).
    [CrossRef]
  14. N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.
  15. Nonpropagating waves also occur in conjunction with a wide-angle beam propagation method discussed in M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]

1991 (4)

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

R. P. Ratowsky, J. A. Fleck, “Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction,” Opt. Lett. 16, 787–789 (1991).
[CrossRef] [PubMed]

W. P. Huang, H. A. Haus, “A simple variational approach to optical rib waveguides,” IEEE J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

The method of lines also works by matrix diagonalization, although the matrices are not so large as those envisaged here. See J. Gerdes, R. Pregla, “Beam propagation algorithm based on the method of lines,” J. Opt. Soc. Am. B 8, 389–395 (1991).
[CrossRef]

1990 (1)

1989 (1)

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

1988 (3)

Nonpropagating waves also occur in conjunction with a wide-angle beam propagation method discussed in M. D. Feit, J. A. Fleck, “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles,”IEE Proc. Part J 135, 56–63 (1988); M. S. Stern, “Semivectorial polarised Hfield solutions for dielectric waveguides with arbitrary index profiles,” Proc. Inst. Electr. Eng. Part J. 135, 333–338 (1988).

N. Dagli, C. Fonstad, “Universal design curves for rib waveguides,” IEEE J. Lightwave Technol. 6, 1136–1145 (1988).
[CrossRef]

1987 (1)

See, for example, W. F. Ames, D. Lee, “Current development in numerical treatment of ocean acoustic propagation,” Appl. Num. Math. 3, 25–47 (1987).
[CrossRef]

1986 (1)

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,”J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

1985 (1)

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).

1984 (1)

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,”IEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984); “Vector H finite element solutions of GaAs/GaAIAs rib waveguides,” Proc. Inst. Electr. Eng. Part J 132, 349–353 (1985).
[CrossRef]

1980 (1)

Ames, W. F.

See, for example, W. F. Ames, D. Lee, “Current development in numerical treatment of ocean acoustic propagation,” Appl. Num. Math. 3, 25–47 (1987).
[CrossRef]

Bisseling, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Cerjan, C.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Dagli, N.

N. Dagli, C. Fonstad, “Universal design curves for rib waveguides,” IEEE J. Lightwave Technol. 6, 1136–1145 (1988).
[CrossRef]

Davies, J. B.

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,”IEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984); “Vector H finite element solutions of GaAs/GaAIAs rib waveguides,” Proc. Inst. Electr. Eng. Part J 132, 349–353 (1985).
[CrossRef]

Dayan, P.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).

Feit, M. D.

Fleck, J. A.

Fonstad, C.

N. Dagli, C. Fonstad, “Universal design curves for rib waveguides,” IEEE J. Lightwave Technol. 6, 1136–1145 (1988).
[CrossRef]

Friesner, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Gerdes, J.

Guldberg, A.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Hammerich, A.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Haus, H. A.

W. P. Huang, H. A. Haus, “A simple variational approach to optical rib waveguides,” IEEE J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Huang, W. P.

W. P. Huang, H. A. Haus, “A simple variational approach to optical rib waveguides,” IEEE J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

Jolicard, G.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Karrlein, W.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Kosloff, R.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Lee, D.

See, for example, W. F. Ames, D. Lee, “Current development in numerical treatment of ocean acoustic propagation,” Appl. Num. Math. 3, 25–47 (1987).
[CrossRef]

LeForestier, C.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Light, J. C.

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,”J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

Lipkin, N.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Meyer, H. D.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Park, T. J.

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,”J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

Pregla, R.

Rahman, B. M. A.

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,”IEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984); “Vector H finite element solutions of GaAs/GaAIAs rib waveguides,” Proc. Inst. Electr. Eng. Part J 132, 349–353 (1985).
[CrossRef]

Ratowsky, R. P.

Ritchie, S.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).

Robertson, M. J.

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).

Roncero, O.

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

Sehmi, N. S.

N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.

Stern, M. S.

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles,”IEE Proc. Part J 135, 56–63 (1988); M. S. Stern, “Semivectorial polarised Hfield solutions for dielectric waveguides with arbitrary index profiles,” Proc. Inst. Electr. Eng. Part J. 135, 333–338 (1988).

Yevick, D.

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

Appl. Num. Math. (1)

See, for example, W. F. Ames, D. Lee, “Current development in numerical treatment of ocean acoustic propagation,” Appl. Num. Math. 3, 25–47 (1987).
[CrossRef]

Appl. Opt. (1)

IEE Proc. Part J (1)

M. S. Stern, “Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles,”IEE Proc. Part J 135, 56–63 (1988); M. S. Stern, “Semivectorial polarised Hfield solutions for dielectric waveguides with arbitrary index profiles,” Proc. Inst. Electr. Eng. Part J. 135, 333–338 (1988).

IEE Trans. Microwave Theory Tech. (1)

B. M. A. Rahman, J. B. Davies, “Finite element analysis of optical and microwave waveguide problems,”IEE Trans. Microwave Theory Tech. MTT-32, 20–28 (1984); “Vector H finite element solutions of GaAs/GaAIAs rib waveguides,” Proc. Inst. Electr. Eng. Part J 132, 349–353 (1985).
[CrossRef]

IEEE J. Lightwave Technol. (2)

W. P. Huang, H. A. Haus, “A simple variational approach to optical rib waveguides,” IEEE J. Lightwave Technol. 9, 56–61 (1991).
[CrossRef]

N. Dagli, C. Fonstad, “Universal design curves for rib waveguides,” IEEE J. Lightwave Technol. 6, 1136–1145 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Yevick, B. Hermansson, “New formulations of the matrix beam propagation method: application to rib waveguides,” IEEE J. Quantum Electron. 25, 221–229 (1989).
[CrossRef]

J. Chem. Phys. (1)

T. J. Park, J. C. Light, “Unitary quantum time evolution by iterative Lanczos reduction,”J. Chem. Phys. 85, 5870–5876 (1986).
[CrossRef]

J. Comput. Phys. (1)

C. LeForestier, R. Bisseling, C. Cerjan, M. D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H. D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, “A comparison of different propagation schemes for the time dependent Schrödinger equation,”J. Comput. Phys. 94, 59–80 (1991).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

Opt. Lett. (1)

Proc. Inst. Electr. Eng. Part J (1)

M. J. Robertson, S. Ritchie, P. Dayan, “Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers,” Proc. Inst. Electr. Eng. Part J 132, 336–342 (1985).

Other (1)

N. S. Sehmi, Large Order Structural Eigenanalysis for Finite Element Systems (Halsted, New York, 1989), pp. 48–69.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of rib waveguide geometry.

Fig. 2
Fig. 2

Evolution of isointensity contours following launch of 0.6-μm-diameter Gaussian beam into structure 1 waveguide.

Fig. 3
Fig. 3

Comparison between Helmholtz (left-hand column) and paraxial (right-hand column) beam propagation for evolution of a beam following injection of overfilling Gaussian beam into structure 1 rib waveguide.

Fig. 4
Fig. 4

Energy transfer between waveguides formed from structure 2 waveguides. Mode of single waveguide was launched into left arm at z = 0. Distances are measured in coupling lengths Lc.

Fig. 5
Fig. 5

Isointensity contours for bound mode of structure 2 waveguide, determined by iteration technique.

Fig. 6
Fig. 6

Isointensity contours for bound mode of structure 3 waveguide, determined by iteration technique.

Tables (5)

Tables Icon

Table 1 Rib Waveguide Parameters

Tables Icon

Table 2 Grid Parameters and Modal Indices

Tables Icon

Table 3 Normalized Propagation Constants, b = (N2N22)/(n22n12), Computed by Different Methods

Tables Icon

Table 4 Coupler Grid Parameters and Coupling Lengths

Tables Icon

Table 5 Normalized Propagation Constants and Coupling Lengths Computed by Different Methods

Equations (37)

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2 Ψ x 2 + 2 Ψ y 2 + 2 Ψ z 2 + n 2 ( x , y ) ω 2 c 2 Ψ = 0.
- 1 2 k 2 Ψ z 2 + i Ψ z = 1 2 k 2 ψ + k 2 [ n 2 ( x , y ) n 0 2 - 1 ] ψ .
i ψ z = 1 2 k 2 ψ + k 2 [ n 2 ( x , y ) n 0 2 - 1 ] ψ .
ψ ( x , y , z ) = ψ n ( x , y ) exp ( - i β n z ) ,
ψ ( x , y , z ) = ψ n ( x , y ) exp ( - i β n z ) ,
ψ n ( x , y ) = ψ n ( x , y ) ,
β n = - k [ 1 - ( 1 + 2 β n / k ) 1 / 2 ] .
- 1 2 k 2 ψ z 2 + i ψ z = H ψ ,
i ψ z = H ψ ,
H = 1 2 k 2 + k 2 [ n 2 ( x , y ) n 0 2 - 1 ]
ψ ( x , y , z ) = n = 0 N A n ψ n ( x , y ) exp { i k [ 1 - ( 1 + 2 β n / k ) 1 / 2 ] z } ,
ψ ( x , y , z ) = n = 0 N A n ψ n ( x , y ) exp ( - i β n z ) ,
u m n ( x , y ) = 1 L 2 exp [ 2 π i L ( m x + n y ) ] , - N 2 < m N 2 ,             - N 2 < n N 2 .
ψ ( x , y , z ) = m = - N / 2 + 1 N / 2 n = - N / 2 + 1 N / 2 B m n ( z ) exp [ 2 π i L ( m x + n y ) ] ,
- 1 2 k 2 ψ z 2 + i ψ z = H ψ ,
i ψ z = H ψ .
- 1 2 k 2 U ψ z 2 + i U ψ z = ( UHU - 1 ) U ψ ,
i U ψ z = ( UHU - 1 ) U ψ .
U ψ ( z ) = exp { i k [ 1 - ( 1 + 2 β / k ) 1 / 2 ] z } U ψ ( 0 ) ,
U ψ ( z ) = exp ( - i β z ) U ψ ( 0 ) ,
ψ ( z ) = U - 1 exp { i k [ 1 - ( 1 + 2 β / k ) 1 / 2 ] z } U ψ ( 0 ) ,
ψ ( z ) = U - 1 exp ( - i β z ) U ψ ( 0 ) .
ψ ( Δ z ) = exp ( - i Δ z H ) ψ ( 0 ) = n = 0 N 1 n ! ( - i Δ z H ) n ψ ( 0 ) + O [ ( Δ z ) N + 1 ] .
q 0 = ψ ( 0 ) , H q 0 = α 0 q 0 + β ¯ 0 q 1 , H q n = β ¯ n - 1 q n - 1 + α n q n + β ¯ n q n + 1 , α n = q n H q n , β ¯ n - 1 = q n - 1 H q n ,
H N = [ α 0 β ¯ 0 0 · 0 0 β ¯ 0 α 1 β ¯ 1 · 0 0 0 β ¯ 1 α 2 · 0 0 · · · · · · 0 0 0 · α N - 1 β ¯ N - 1 0 0 0 · β ¯ N - 1 α N ] .
ψ ( Δ z ) = U - 1 exp { i k [ 1 - ( 1 + 2 β N / k ) 1 / 2 ] Δ z } U ψ ( 0 ) ,
ψ ( Δ z ) = U - 1 exp ( - i β N Δ z ) U ψ ( 0 ) ,
Δ z < π 4 min | 1 Re { k [ 1 - ( 1 + 2 β n / k ) 1 / 2 ] } | ,
Δ z < π 4 min | 1 β n | ,
2 β n / k < - 1 ,
2 β n / k - ( κ x 2 + κ y 2 ) / k 2 ,
θ = sin - 1 ( κ x 2 + κ y 2 ) 1 / 2 / k ,
β 0 , β 1 , , β N ,
β 0 > β 2 > > β N .
U = [ u 0 u 1 u N ] .
q 0 m + 1 = n = 0 N [ u 0 ] n q n m ,
| β 0 m + 1 - β 0 m β 0 m | < .

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