Abstract

Optical soliton pulses offer many applications within optical communication systems, but by definition a soliton is only subjected to second-order anomalous group-velocity-dispersion; an understanding of higher-order dispersion is necessary for practical implementation of soliton pulses. A numerical model of a waveguide was developed using the nonlinear Schrödinger equation, with parameters set to ensure the input pulse energy would be equal to the fundamental soliton energy. Higher-order group-velocity-dispersion was gradually increased, for various temporal widths and waveguide dispersions. A minimum pulse duration of 100 fs was determined to be necessary for fundamental soliton pulse propagation in practical photonic crystal waveguides.

© 2013 Optical Society of America

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References

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  1. G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  2. B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  3. P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).
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    [CrossRef]
  5. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
    [CrossRef]
  6. A. C. Peacock, “Soliton propagation in tapered silicon core fibers,” Opt. Lett. 35, 3697–3699 (2010).
    [CrossRef]
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    [CrossRef]
  9. J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  10. S. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).
  11. N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).
  12. W. Ding, C. Benton, A. V. Gorbach, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. Gnan, M. Sorrel, and R. M. De La Rue, “Solitons and spectral broadening in long silicon-on-insulator photonic wires,” Opt. Express 16, 3310–3319 (2008).
    [CrossRef]
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  14. T. C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).
  15. C. Husko, A. De Rossi, and C. W. Wong, “Effect of multi-photon absorption and free carriers on self-phase modulation in slow-light photonic crystals,” Opt. Lett. 36, 2239–2241 (2011).
    [CrossRef]
  16. C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
    [CrossRef]
  17. W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).
  18. Q. Lin, O. Painter, and G. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
  19. D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).
  20. W. Ding, A. V. Gorbach, W. J. Wadswarth, J. C. Knight, D. V. Skryabin, M. J. Strain, M. Sorel, and R. M. De La Rue, “Time and frequency domain measurements of solitons in subwavelength silicon waveguides using a cross-correlation technique,” Opt. Express 18, 26625–26630 (2010).
    [CrossRef]
  21. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. 31, 1295–1297 (2006).
    [CrossRef]
  22. C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).
    [CrossRef]
  23. J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
    [CrossRef]
  24. M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12, 1551–1561 (2004).
    [CrossRef]
  25. R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).
  26. R. Haberman, Applied Partial Differential Equations, with Fourier Series and Boundary Value Problems (Pearson Prentice-Hall, 2004).
  27. R. Paschotta, “Transform limit,” http://www.rp-photonics.com/transform_limit.html .
  28. R. Paschotta, “Timebandwidth product,” http://www.rp-photonics.com/time_bandwidth_product.html .
  29. G. Strang, Linear Algebra and its Applications, 3rd ed. (Brooks/Cole, 1988).

2011 (3)

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).

D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).

C. Husko, A. De Rossi, and C. W. Wong, “Effect of multi-photon absorption and free carriers on self-phase modulation in slow-light photonic crystals,” Opt. Lett. 36, 2239–2241 (2011).
[CrossRef]

2010 (4)

2009 (1)

2008 (1)

2007 (4)

2006 (1)

2005 (1)

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
[CrossRef]

2004 (1)

2002 (1)

C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
[CrossRef]

Agrawal, G.

Agrawal, G. P.

Benton, C.

Benton, C. J.

Boyd, R. W.

Chen, C.-F.

C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
[CrossRef]

Chen, X.

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

Chi, S.

C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
[CrossRef]

Colman, P.

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Combrie, S.

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Dadap, J.

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

De La Rue, R. M.

De Rossi, A.

C. Husko, A. De Rossi, and C. W. Wong, “Effect of multi-photon absorption and free carriers on self-phase modulation in slow-light photonic crystals,” Opt. Lett. 36, 2239–2241 (2011).
[CrossRef]

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Ding, W.

Eggleton, S.

S. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).

Fainman, Y.

D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).

Fauchet, P. M.

Gnan, M.

Gorbach, A. V.

Haberman, R.

R. Haberman, Applied Partial Differential Equations, with Fourier Series and Boundary Value Problems (Pearson Prentice-Hall, 2004).

Hseih, W.

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

Husko, C.

C. Husko, A. De Rossi, and C. W. Wong, “Effect of multi-photon absorption and free carriers on self-phase modulation in slow-light photonic crystals,” Opt. Lett. 36, 2239–2241 (2011).
[CrossRef]

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Joannopoulos, J.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kent, E.

R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).

Kim, T.

T. C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).

Kinsler, P.

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2004).

Knight, J. C.

Kuramochi, E.

Kwong, D.-L.

Lin, Q.

Luo, B.

C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
[CrossRef]

McMillan, J. F.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observations of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express 18, 15484–15497 (2010).
[CrossRef]

Meade, R.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Mitsugi, S.

Nagle, R.

R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).

New, G. H. C.

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
[CrossRef]

Notomi, M.

Osgood, R.

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

Painter, O.

Panoiu, N.

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

Panoiu, N. C.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).

Peacock, A. C.

Piredda, G.

Poon, T. C.

T. C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).

Ryu, H.

Sadiku, M.

M. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. (CRC Press, 2001).

Saff, B.

R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).

Sagnes, I.

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Shinya, A.

Skryabin, D. V.

Snider, A. D.

R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).

Sorel, M.

Sorrel, M.

Strain, M. J.

Strang, G.

G. Strang, Linear Algebra and its Applications, 3rd ed. (Brooks/Cole, 1988).

Sun, P.

D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).

Tan, D.

D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tyrrell, J. C. A.

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
[CrossRef]

Wadswarth, W. J.

Wadsworth, W. J.

Winn, J.

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Wong, C. W.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).

C. Husko, A. De Rossi, and C. W. Wong, “Effect of multi-photon absorption and free carriers on self-phase modulation in slow-light photonic crystals,” Opt. Lett. 36, 2239–2241 (2011).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observations of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express 18, 15484–15497 (2010).
[CrossRef]

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Yin, L.

Yu, M.

Zhang, J.

Appl. Phys. A (1)

N. C. Panoiu, J. F. McMillan, and C. W. Wong, “Influence of the group-velocity on the pulse propagation in 1D silicon photonic crystal waveguides,” Appl. Phys. A 103, 835–838 (2011).

J. Mod. Opt. (1)

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt. 52, 973 (2005).
[CrossRef]

Nat. Commun. (1)

D. Tan, P. Sun, and Y. Fainman, “Monolithic nonlinear pulse compressor on a silicon chip,” Nat. Commun. 1, 116 (2011).

Nat. Photonics (1)

P. Colman, C. Husko, S. Combrie, I. Sagnes, C. W. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics 21, 862–868 (2010).

Opt. Express (8)

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H. Ryu, “Waveguides, resonators and their coupled elements in photonic crystal slabs,” Opt. Express 12, 1551–1561 (2004).
[CrossRef]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007).
[CrossRef]

Q. Lin, O. Painter, and G. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).

W. Ding, C. Benton, A. V. Gorbach, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. Gnan, M. Sorrel, and R. M. De La Rue, “Solitons and spectral broadening in long silicon-on-insulator photonic wires,” Opt. Express 16, 3310–3319 (2008).
[CrossRef]

C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express 17, 5879–5884 (2009).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observations of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express 18, 15484–15497 (2010).
[CrossRef]

W. Hseih, X. Chen, J. Dadap, N. Panoiu, and R. Osgood, “Cross-phase modulation-induced spectral and teporal effects on co-propagating femtosecond pulses in silicon photonic wires,” Opt. Express 15, 3310–3319 (2007).

W. Ding, A. V. Gorbach, W. J. Wadswarth, J. C. Knight, D. V. Skryabin, M. J. Strain, M. Sorel, and R. M. De La Rue, “Time and frequency domain measurements of solitons in subwavelength silicon waveguides using a cross-correlation technique,” Opt. Express 18, 26625–26630 (2010).
[CrossRef]

Opt. Lett. (4)

Optik (1)

C.-F. Chen, S. Chi, and B. Luo, “Femtosecond soliton propagation in an optical fiber,” Optik 113, 267–271 (2002).
[CrossRef]

Other (12)

R. Nagle, E. Kent, B. Saff, and A. D. Snider, Fundamentals of Differential Equations and Boundary Value Problems, 3rd ed. (Addison Wesley, 2000).

R. Haberman, Applied Partial Differential Equations, with Fourier Series and Boundary Value Problems (Pearson Prentice-Hall, 2004).

R. Paschotta, “Transform limit,” http://www.rp-photonics.com/transform_limit.html .

R. Paschotta, “Timebandwidth product,” http://www.rp-photonics.com/time_bandwidth_product.html .

G. Strang, Linear Algebra and its Applications, 3rd ed. (Brooks/Cole, 1988).

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2004).

J. Joannopoulos, S. Johnson, J. Winn, and R. Meade, Photonic Crystals, Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

S. Eggleton, Nonlinear Photonic Crystals (Springer, 2003).

M. Sadiku, Numerical Techniques in Electromagnetics, 2nd ed. (CRC Press, 2001).

T. C. Poon and T. Kim, Engineering Optics with Matlab (World Scientific, 2006).

G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

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Figures (4)

Fig. 1.
Fig. 1.

Ideal soliton propagation for the (a) fundamental soliton N=1, (b) higher-order soliton N=2, and (c) higher-order soliton N=3.

Fig. 2.
Fig. 2.

Maximum third-order GVD coefficients for a 2 fs pulse, as a function of nonlinear length and second-order GVD, obtained through (a) direct NLSE simulations and (b) Eq. (18).

Fig. 3.
Fig. 3.

Output TBP over the input TBP for an optical pulse propagating 10 nonlinear lengths, with an input pulse energy equal to that of the fundamental soliton, and with temporal durations ranging from 1 fs to 100 ps. The TOD has a value of (a) 0ps3/mm and (b) 0.1ps3/mm.

Fig. 4.
Fig. 4.

Data plots of soliton disturbance as a function of TOD for 10 nonlinear lengths, both for (a) TBP and (b) phase.

Tables (1)

Tables Icon

Table 1. Coefficients for Eq. (18)

Equations (18)

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

β(ω)=n(ω)ωc0,
βM(ω)=dMβ(ω)dωM,
β1(ω)=1vg=ngc0=1c0(n(ω)+ωdn(ω)dω),
β3(ω)=dβ2(ω)dω=d2β1(ω)dω2=d3β(ω)dω3,
Δn=n2I1=3η0χ(3)ϵ0nI1,
Az+1vgAt+j2β22At2+j6β33At3=jγ|A|2A·exp(αz),
γ=2πn2Ae·λ0·(ngn)2,
A¯(z,ω)=A¯(0,ω)·exp[i12β2ω2z+i16β3ω3z+],
A¯(z,ω)=A(z,T)·exp[2πiωT]dT,
T=tzvg,
A(z,T)=A(0,T)·exp[i(2πλ0)·n2·|A(0,T)|2].
EP=2K|β2||γ|·τ,
LNL=L/τ2|β2|,
A^x¯=b¯,
A^TA^x¯=A^Tb¯,
(A^TA^)1(A^TA^)x¯=x¯=(A^TA^)1A^Tb¯,
f(τ,β2,LNL)=C1·f1(τ,β2,LNL)+C2·f2(τ,β2,LNL)+C3,
f(τ,β2,LNL)=C1·τ·exp(LNL)+C2·β2·exp(LNL)+C3,

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