Abstract

A method that uses discrete wavelet transforms for the solution of evolution equations that describe optical pulse propagation in nonlinear media is presented. The theory of orthogonal wavelet transforms is outlined and applied to the representation of optical pulses. Wavelet transform representations of propagation operators are presented and applied to the nonlinear Schrödinger equation, yielding results that are indistinguishable from traditional Fourier-based simulations. The compression properties of wavelet representations of optical pulses permit significant improvement in execution speed compared with that of the split-step Fourier method.

© 2000 Optical Society of America

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References

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  1. M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge U. Press, Cambridge, UK, 1991).
  2. T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Physiol. 55, 203–230 (1984).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed., Optics and Photonics Series (Academic, Orlando, Fla., 1995).
  4. R. W. Ramirez, The FFT, Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).
  5. C. Van Loan, Computational Frameworks for the Fast Fourier Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  6. M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
    [CrossRef]
  7. Ref. 3, p. 51.
  8. D. Yevick, B. Hermansson, “New fast Fourier transform and finite element approaches to the calculation of multiple-stripe-geometry laser modes,” J. Appl. Phys. 59, 1769–1771 (1986).
    [CrossRef]
  9. H. J. A. da Silva, J. J. O’Reilly, “Optical pulse modeling with Hermite–Gaussian functions,” Opt. Lett. 14, 526–528 (1989).
    [CrossRef] [PubMed]
  10. G. Strang, T. Q. Nguyen, Wavelets and Filter Banks (Wellesley Cambridge Press, Wellesley, Mass., 1996).
  11. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Mass., 1992).
  12. J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
    [CrossRef] [PubMed]
  13. G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  15. G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,” SIAM J. Numer. Anal. 29, 1716–1740 (1992).
    [CrossRef]
  16. B. Jawerth, W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev. 36, 377–412 (1994).
    [CrossRef]
  17. P. Charton, V. Perrier, “Rapid matrix-vector products using wavelet transform—application to numerical-solution of partial-differential equations,” Model. Math. Anal. Numer. 29, 701–747 (1995).

1995 (2)

J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

P. Charton, V. Perrier, “Rapid matrix-vector products using wavelet transform—application to numerical-solution of partial-differential equations,” Model. Math. Anal. Numer. 29, 701–747 (1995).

1994 (2)

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

B. Jawerth, W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev. 36, 377–412 (1994).
[CrossRef]

1992 (1)

G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,” SIAM J. Numer. Anal. 29, 1716–1740 (1992).
[CrossRef]

1991 (1)

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

1989 (1)

1986 (1)

D. Yevick, B. Hermansson, “New fast Fourier transform and finite element approaches to the calculation of multiple-stripe-geometry laser modes,” J. Appl. Phys. 59, 1769–1771 (1986).
[CrossRef]

1984 (1)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Physiol. 55, 203–230 (1984).
[CrossRef]

Ablowitz, M. J.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Physiol. 55, 203–230 (1984).
[CrossRef]

M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge U. Press, Cambridge, UK, 1991).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed., Optics and Photonics Series (Academic, Orlando, Fla., 1995).

Belzer, B.

J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

Beylkin, G.

G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,” SIAM J. Numer. Anal. 29, 1716–1740 (1992).
[CrossRef]

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Chang, Y. H.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Charton, P.

P. Charton, V. Perrier, “Rapid matrix-vector products using wavelet transform—application to numerical-solution of partial-differential equations,” Model. Math. Anal. Numer. 29, 701–747 (1995).

Clarkson, P. A.

M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge U. Press, Cambridge, UK, 1991).

Coifman, R.

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

da Silva, H. J. A.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Mass., 1992).

Delfyett, P. J.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Dienes, A.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Heritage, J. P.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Hermansson, B.

D. Yevick, B. Hermansson, “New fast Fourier transform and finite element approaches to the calculation of multiple-stripe-geometry laser modes,” J. Appl. Phys. 59, 1769–1771 (1986).
[CrossRef]

Hong, M. Y.

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

Jawerth, B.

B. Jawerth, W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev. 36, 377–412 (1994).
[CrossRef]

Liao, J.

J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

Nguyen, T. Q.

G. Strang, T. Q. Nguyen, Wavelets and Filter Banks (Wellesley Cambridge Press, Wellesley, Mass., 1996).

O’Reilly, J. J.

Perrier, V.

P. Charton, V. Perrier, “Rapid matrix-vector products using wavelet transform—application to numerical-solution of partial-differential equations,” Model. Math. Anal. Numer. 29, 701–747 (1995).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Ramirez, R. W.

R. W. Ramirez, The FFT, Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Rokhlin, V.

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

Strang, G.

G. Strang, T. Q. Nguyen, Wavelets and Filter Banks (Wellesley Cambridge Press, Wellesley, Mass., 1996).

Sweldens, W.

B. Jawerth, W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev. 36, 377–412 (1994).
[CrossRef]

Taha, T. R.

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Physiol. 55, 203–230 (1984).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Van Loan, C.

C. Van Loan, Computational Frameworks for the Fast Fourier Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Vetterling, W. H.

W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Villasenor, J. D.

J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

Yevick, D.

D. Yevick, B. Hermansson, “New fast Fourier transform and finite element approaches to the calculation of multiple-stripe-geometry laser modes,” J. Appl. Phys. 59, 1769–1771 (1986).
[CrossRef]

Commun. Pure Appl. Math. (1)

G. Beylkin, R. Coifman, V. Rokhlin, “Fast wavelet transforms and numerical analysis. I,” Commun. Pure Appl. Math. 44, 141–183 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, “Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment,” IEEE J. Quantum Electron. 30, 1122–1131 (1994).
[CrossRef]

IEEE Trans. Image Process. (1)

J. D. Villasenor, B. Belzer, J. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. Image Process. 4, 1053–1060 (1995).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

D. Yevick, B. Hermansson, “New fast Fourier transform and finite element approaches to the calculation of multiple-stripe-geometry laser modes,” J. Appl. Phys. 59, 1769–1771 (1986).
[CrossRef]

J. Comp. Physiol. (1)

T. R. Taha, M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comp. Physiol. 55, 203–230 (1984).
[CrossRef]

Model. Math. Anal. Numer. (1)

P. Charton, V. Perrier, “Rapid matrix-vector products using wavelet transform—application to numerical-solution of partial-differential equations,” Model. Math. Anal. Numer. 29, 701–747 (1995).

Opt. Lett. (1)

SIAM J. Numer. Anal. (1)

G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,” SIAM J. Numer. Anal. 29, 1716–1740 (1992).
[CrossRef]

SIAM Rev. (1)

B. Jawerth, W. Sweldens, “An overview of wavelet based multiresolution analyses,” SIAM Rev. 36, 377–412 (1994).
[CrossRef]

Other (8)

M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge U. Press, Cambridge, UK, 1991).

W. H. Press, S. A. Teukolsky, W. H. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed., Optics and Photonics Series (Academic, Orlando, Fla., 1995).

R. W. Ramirez, The FFT, Fundamentals and Concepts (Prentice-Hall, Englewood Cliffs, N.J., 1985).

C. Van Loan, Computational Frameworks for the Fast Fourier Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

G. Strang, T. Q. Nguyen, Wavelets and Filter Banks (Wellesley Cambridge Press, Wellesley, Mass., 1996).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Mass., 1992).

Ref. 3, p. 51.

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Figures (4)

Fig. 1
Fig. 1

D6 wavelet.

Fig. 2
Fig. 2

D6 scaling function.

Fig. 3
Fig. 3

Second-order soliton propagation in the absence of loss.

Fig. 4
Fig. 4

Gaussian pulse propagating in a lossy, anomalous-dispersion regime.

Tables (3)

Tables Icon

Table 1 Derivative Operator Coefficients for the D6 Scaling Function

Tables Icon

Table 2 Second-Derivative Coefficients

Tables Icon

Table 3 Parameters for Standard Silica Communication Fiber

Equations (69)

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

Az=-i2β22AT2-16β33AT3-α2A+iγ|A|2A.
zA(T, z)=(Lˆ+Nˆ)A(T, z),
Lˆ=-i2β22T2-16β33T3-α2,
Nˆ=iγ|A|2.
A(T, z+h)=expzz+h(Nˆ+Lˆ)dζA(T, z)
exph2Lˆexpzz+hNˆdζ×exph2LˆA(T, z).
zz+hNˆdζ(h/2)[Nˆ(z)+Nˆ(z+h)].
ψn,k(t)=2n/2ψ(2nt-k).
f0(t)=kν0, kϕ(t-k),
ν0,k=-f(t)ϕ(t-k)dt.
fn(t)=kνn,kϕn,k(t),
ϕn,k(t)=2n/2ϕ(2nt-k).
-ϕn,jϕn,kdt=δj,k,
δj,k=1,j=k0jk.
V0V1V2VnL2(R).
Vn=Vn-1Wn-1,
Vn=V0W0W1Wn-2Wn-1.
-ψn,kψm, j dt=δm,nδj,k.
-ϕn,kψn,j dt=0.
1.VnVn+1, Vn={0},Vn¯=L2(R),
2.ϕ(t)V0  ϕ(t-k)V0,
3.ϕ(t)V0  ϕ(2t)V1.
ϕ(t)=21/2k=0Lhkϕ(2t-k).
ψ(t)=21/2k=0Lgkϕ(2t-k).
-ϕ(t)dt=1
khk=2.
f(t)fN(t)=kνN,kϕN,k(t).
f(t)fN(t)=kν0, kϕ0,k(t)+j=0N-1kμj,kψj,k(t).
νn,k=-fn(t)ϕn,k(t)dt.
fn(t)=kνn-1,kϕn-1,k(t)+kμn-1,kψn-1,k(t),
νn,k=-lνn-1,lϕn-1,l+lμn-1,lψn-1,lϕn,kdt.
ϕn-1,l=php-2lϕn,p,
ψn-1,l=pgp-2lϕn,p
νn,k=lνn-1,lphp-2l-ϕn,pϕn,kdt+lμn-1,lpgp-2l-ϕn,pϕn,kdt,
νn,k=lhk-2lνn-1,l+lgk-2lμn-1,l.
νn-1,k=-fn(t)ϕn-1,kdt,
νn-1,k=-lνn,lϕn,lϕn-1,kdt.
νn-1,k=lνn,lphp-2k-ϕn,lϕn,pdt.
νn-1,k=lhl-2kνn,l,
μn-1,k=lgl-2kνn,l.
f(t)fN(t)=kνN, kϕN, k(t).
fn=f0+w0+w1++wN-1,
f0=kν0,kϕ0,k,
w0=kμ0,kψ0,k,
wN-1=kμN, kψN, k.
A(T, z+(h/2))=exp[(h/2)L]A(T, z).
A(T, z)=k=0j-1νj,kϕj,k(t).
νj,kA(2-jk, z).
TA(T, z)=k=02j-1νj,kϕj,k(t),
νj,k=l=02j-1rk,l jνj,l,
νj=Rjνj,
rk,lj=2j-ϕ(2jt-k)ϕ(2jt-l)2jdt,
rk,lj=2jrk-l,
rλ=-ϕ(t-λ)ddtϕ(t)dt.
-L+1λL-1,
rλ=-r-λ.
rλ=r-λ.
rk,lj=rk-l,
rλ=-ϕ(t-λ)dndtnϕ(t)dt
μa=Wa=W M b,
μa=Wa=W M Ib,
IW-1W,
μa=Wa=W M W-1·Wb.
W-1WT or WTW=I.
μa=Wa=WMWT·Wb.
aj,k=a(k-2lj)
aj,k=a(2lk-j),
exph2Lˆ1+h2Lˆ+h28Lˆ2+h348Lˆ3.

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