Abstract

An efficient method for the design of optical devices based on codirectional grating-assisted mode coupling is presented. A low-complexity algorithm is developed to calculate the coupling function of a grating that accurately matches an arbitrarily given target spectral response. The method relies on the synthesis of the grating impulse response by means of an exact differential layer-peeling algorithm.

© 2000 Optical Society of America

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References

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  1. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
    [CrossRef]
  2. D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
    [CrossRef]
  3. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  4. G. Meltz, W. W. Morey, W. H. Glen, “Formation of Bragg gratings in optical fibres by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  5. A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalisers,” Opt. Lett. 21, 336–338 (1996).
    [CrossRef] [PubMed]
  6. P. St. J. Russell, D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
    [CrossRef]
  7. J. N. Blake, B. Y. Kim, H. E. Engan, H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12, 281–283 (1987).
    [CrossRef] [PubMed]
  8. K. I. Hopcraft, P. R. Smith, An Introduction to Electromagnetic Inverse Scattering (Kluwer Academic, Dordrecht, The Netherlands, 1992).
  9. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  10. B. G. Bovard, “Fourier transform technique applied to quarterwave optical coatings,” Appl. Opt. 27, 3062–3063 (1988).
    [CrossRef] [PubMed]
  11. K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
    [CrossRef]
  12. I. M. Gel’fand, B. M. Levitan, “On a determination of a differential equation from its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).
  13. I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
    [CrossRef]
  14. G. H. Song, S. Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  15. J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
    [CrossRef]
  16. E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
    [CrossRef]
  17. B. E. Little, C. Wu, W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. 20, 1259–1261 (1995).
    [CrossRef] [PubMed]
  18. J. Skaar, K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
    [CrossRef]
  19. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
    [CrossRef]
  20. A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29, 359–389 (1987).
    [CrossRef]
  21. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  22. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  23. J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988).
  24. E. M. Dowling, D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471–486 (1994).
    [CrossRef]
  25. G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980), pp. 107–112.
  26. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), pp. 104–146.
  27. A. B. Carlson, Communication Systems (McGraw-Hill, Singapore, 1986), pp. 103–105.
  28. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 92–108 and 384–388.

1999

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998

1996

A. M. Vengsarkar, J. R. Pedrazzani, J. B. Judkins, P. J. Lemaire, N. S. Bergano, C. R. Davidson, “Long-period fiber-grating-based gain equalisers,” Opt. Lett. 21, 336–338 (1996).
[CrossRef] [PubMed]

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

1995

B. E. Little, C. Wu, W. P. Huang, “Synthesis of codirectional couplers with ultralow sidelobes and minimum bandwidth,” Opt. Lett. 20, 1259–1261 (1995).
[CrossRef] [PubMed]

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

1994

E. M. Dowling, D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471–486 (1994).
[CrossRef]

W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

1993

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

1990

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

P. St. J. Russell, D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

1989

1988

1987

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

J. N. Blake, B. Y. Kim, H. E. Engan, H. J. Shaw, “Analysis of intermodal coupling in a two-mode fiber with periodic microbends,” Opt. Lett. 12, 281–283 (1987).
[CrossRef] [PubMed]

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29, 359–389 (1987).
[CrossRef]

1985

1978

1973

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

1972

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1960

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

1955

I. M. Gel’fand, B. M. Levitan, “On a determination of a differential equation from its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), pp. 104–146.

Bergano, N. S.

Blake, J. N.

Bovard, B. G.

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29, 359–389 (1987).
[CrossRef]

Capmany, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Carlson, A. B.

A. B. Carlson, Communication Systems (McGraw-Hill, Singapore, 1986), pp. 103–105.

Davidson, C. R.

Dobrowolski, J. A.

Dowling, E. M.

E. M. Dowling, D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471–486 (1994).
[CrossRef]

Engan, H. E.

Feced, R.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On a determination of a differential equation from its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Glen, W. H.

Hand, D. P.

P. St. J. Russell, D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

Hopcraft, K. I.

K. I. Hopcraft, P. R. Smith, An Introduction to Electromagnetic Inverse Scattering (Kluwer Academic, Dordrecht, The Netherlands, 1992).

Huang, W. P.

Judkins, J. B.

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29, 359–389 (1987).
[CrossRef]

Kay, I.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

Kim, B. Y.

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980), pp. 107–112.

Lemaire, P. J.

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On a determination of a differential equation from its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

Little, B. E.

Lowe, D.

MacFarlane, D. L.

E. M. Dowling, D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471–486 (1994).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 92–108 and 384–388.

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988).

Marcuse, D.

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

Marti, J.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Meltz, G.

Morey, W. W.

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Pedrazzani, J. R.

Peral, E.

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988).

Risvik, K. M.

Roman, J. E.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Shaw, H. J.

Shin, S. Y.

Skaar, J.

Smith, P. R.

K. I. Hopcraft, P. R. Smith, An Introduction to Electromagnetic Inverse Scattering (Kluwer Academic, Dordrecht, The Netherlands, 1992).

Song, G. H.

Vengsarkar, A. M.

Winick, K. A.

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 92–108 and 384–388.

Wu, C.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

Zervas, M. N.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Am. Math. Soc. Trans.

I. M. Gel’fand, B. M. Levitan, “On a determination of a differential equation from its spectral function,” Am. Math. Soc. Trans. 1, 253–304 (1955).

Appl. Opt.

Commun. Pure Appl. Math.

I. Kay, “The inverse scattering problem when the reflection coefficient is a rational function,” Commun. Pure Appl. Math. 13, 371–393 (1960).
[CrossRef]

Electron. Lett.

P. St. J. Russell, D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846–1848 (1990).
[CrossRef]

IEEE J. Quantum Electron.

K. A. Winick, J. E. Roman, “Design of corrugated waveguide filters by Fourier-transform techniques,” IEEE J. Quantum Electron. 26, 1918–1929 (1990).
[CrossRef]

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

J. E. Roman, K. A. Winick, “Waveguide grating filters for dispersion compensation and pulse compression,” IEEE J. Quantum Electron. 29, 975–982 (1993).
[CrossRef]

E. Peral, J. Capmany, J. Marti, “Iterative solution to the Gel’fand–Levitan–Marchenko coupled equations and application to synthesis of fiber gratings,” IEEE J. Quantum Electron. 32, 2078–2084 (1996).
[CrossRef]

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

J. Appl. Phys.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol.

J. Skaar, K. M. Risvik, “A genetic algorithm for the inverse problem in synthesis of fiber gratings,” J. Lightwave Technol. 16, 1928–1932 (1998).
[CrossRef]

D. Marcuse, “Directional couplers made of nonidentical asymmetric slabs. Part II: grating assisted couplers,” J. Lightwave Technol. 5, 268–273 (1987).
[CrossRef]

E. M. Dowling, D. L. MacFarlane, “Lightwave lattice filters for optically multiplexed communication systems,” J. Lightwave Technol. 12, 471–486 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 45, 312–335 (1995).
[CrossRef]

SIAM Rev.

A. M. Bruckstein, T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev. 29, 359–389 (1987).
[CrossRef]

Other

J. G. Proakis, D. G. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1988).

K. I. Hopcraft, P. R. Smith, An Introduction to Electromagnetic Inverse Scattering (Kluwer Academic, Dordrecht, The Netherlands, 1992).

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980), pp. 107–112.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), pp. 104–146.

A. B. Carlson, Communication Systems (McGraw-Hill, Singapore, 1986), pp. 103–105.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 92–108 and 384–388.

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Figures (12)

Fig. 1
Fig. 1

Grating-assisted codirectional coupler (GACC). F: fast mode, S: slow mode.

Fig. 2
Fig. 2

(a) Space–time paths corresponding to the propagation of two coupled modes through a grating as described by Eqs. (4) and (b) lattice digital-filter model for the discrete GACC.

Fig. 3
Fig. 3

(a) Space–time paths corresponding to the propagation through the grating of two coupled modes when time delays are measured with respect to the fast mode and (b) corresponding lattice digital-filter model for the discrete GACC.

Fig. 4
Fig. 4

Zero diagram in the complex plane for the TFF(Z) transfer function of a uniform GACC for the following values of the product qL: 10-150/(π/2), 0.5/(π/2), 1/(π/2), 2/(π/2), 3/(π/2), and 8(π/2).

Fig. 5
Fig. 5

Space–time diagram for the equivalent contradirectional problem. #n represents those paths with 2n-1 scattering events.

Fig. 6
Fig. 6

Space–time diagram that illustrates the reconstruction process of the equivalent contradirectional problem for the discretized GACC.

Fig. 7
Fig. 7

Reconstruction of a uniform grating with a q of 0.0283 cm-1 and a length of 50 cm: reconstructed grating (solid curve) and original grating (dotted curve).

Fig. 8
Fig. 8

Undercoupled realization of a uniform overcoupled grating with a q of 0.0471 cm-1 and a length of 50 cm. (a) Spectral response |TSF(β)|2. (b) Coupling function q(z): undercoupled grating (solid curve) and overcoupled grating (dotted curve).

Fig. 9
Fig. 9

Spectral response |TSF(β)|2 of the synthesized top-flat filter in linear and logarithmic scales.

Fig. 10
Fig. 10

Coupling function q(z) of the synthesized top-flat filter.

Fig. 11
Fig. 11

Spectral response |TSF(β)|2 of the synthesized triangular filter in linear and logarithmic scales.

Fig. 12
Fig. 12

Real and imaginary parts of the coupling function q(z) that corresponds to the synthesized triangular filter.

Equations (20)

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

dbS(z, β)dz-jβbS(z, β)=-q*(z)bF(z, β),
dbF(z, β)dz+jβbF(z, β)=q(z)bS(z, β).
β(ω)=βS(ω)-βF(ω)-K02(βS-βF)ωω0ω-ω02
Δneffc π(f-f0),
eS(z, β)=bS(z, β)expj βS(ω)+βF(ω)2 z×exp+j K0z2,
eF(z, β)=bF(z, β)expj βS(ω)+βF(ω)2 z×exp-j K0z2.
bS(z, τ)z+bS(z, τ)τ=-q*(z)bF(z, τ),
bF(z, τ)z-bF(z, τ)τ=q(z)bS(z, τ).
t=βτω-ω0=Δneff2c τ.
bS(L, β)bF(L, β)=TSS(β)TSF(β)TFS(β)TFF(β) bS(0, β)bF(0, β).
TFF(β)=TSS*(β),
TSF(β)=-TFS*(β),
|TSF(β)|2+|TFF(β)|2=1.
(bS(z+Δ,β)bF(z+Δ,β))=[cos(ξΔ)+jβξsin(ξΔ)-q*ξsin(ξΔ)qξsin(ξΔ)cos(ξΔ)-jβξsin(ξΔ)]MT(Δ,β)× (bS(z,β)bF(z,β)),
MT(Δ,β)[exp(jβΔ2)00exp(-jβΔ2)]MP(Δ2,β)× [cos(|q|Δ)-q*|q|sin(|q|Δ)q|q|sin(|q|Δ)cos(|q|Δ)]MC(Δ,q)× [exp(jβΔ2)00exp(-jβΔ2)]MP(Δ2,β).
TFF(Z)1+2n=0NanZ-n,
Heq(β)=TSF(β)TFF(β),
heq(τ)=-12 q*τ2+hT(τ),
Δh(2(N+1)Δ)=-q*((N+1)Δ)2tan[|q((N+1)Δ)|Δ]|q((N+1)Δ)|Δ×m=0N{cos[|q(mΔ)|Δ]}-2,
MT-1((N+1)Δ, β)=MC-1(Δ, q((N+1)Δ))Mp-1(Δ, β)MT-1((N+1)Δ, β),

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