Abstract

We apply our recently developed time-transformation method for studying the propagation of few-cycle optical pulses inside a nonlinear Kerr medium after taking into account that changes in the refractive index vary with the electric field as E2 and not by its average over an optical cycle. Our technique correctly predicts carrier-wave shocking and generation of odd-order harmonics inside a Kerr medium, the two features found earlier with directly solving Maxwell’s equations using the finite-difference time-domain (FDTD) methods. We extend our method to study the impact of a finite response of the Kerr nonlinearity on harmonic generation and to include chromatic dispersion that cannot be ignored for ultrashort pulses. We show that nonlinear effects can help in controlling the width of an ultrashort pulse, even though it cannot propagate as a fundamental soliton. Our time-transformation method provides an alternative to the FDTD technique, as it deals with the electric field directly but does not require step size to be a small fraction of the wavelength, resulting in much faster computation speeds.

© 2013 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).
  2. T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
    [CrossRef]
  3. G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, Opt. Express 15, 5382 (2007).
    [CrossRef]
  4. P. M. Goorjian and A. Taflove, Opt. Lett. 17, 180 (1992).
    [CrossRef]
  5. R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
    [CrossRef]
  6. Y. Xiao, G. P. Agrawal, and D. N. Maywar, Opt. Lett. 37, 1271 (2012).
    [CrossRef]
  7. Y. Xiao, D. N. Maywar, and G. P. Agrawal, J. Opt. Soc. Am. B 29, 2958 (2012).
    [CrossRef]
  8. L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
    [CrossRef]

2012 (2)

2007 (1)

1999 (1)

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

1997 (1)

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

1996 (1)

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef]

1992 (1)

Agrawal, G. P.

Brabec, T.

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Dudley, J. M.

Flesch, R. G.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef]

Genty, G.

Gilles, L.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Goorjian, P. M.

Kibler, B.

Kinsler, P.

Krausz, F.

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Maywar, D. N.

Moloney, J. V.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef]

Pushkarev, A.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef]

Taflove, A.

Vázquez, L.

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Xiao, Y.

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (1)

L. Gilles, J. V. Moloney, and L. Vázquez, Phys. Rev. E 60, 1051 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

R. G. Flesch, A. Pushkarev, and J. V. Moloney, Phys. Rev. Lett. 76, 2488 (1996).
[CrossRef]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Electric fields and (b) optical spectra at the input (dashed blue curves) and output ends of a nondispersive Kerr medium using our new approach (dot-dashed red curves) and the FDTD method (solid yellow curves).

Fig. 2.
Fig. 2.

Changes in the relative amplitudes of the first three harmonics with the Kerr response time τk. In all cases, the amplitude decreases almost exponentially with increasing τk.

Fig. 3.
Fig. 3.

Electric fields of a 10 fs optical pulse after it has propagated for 50 μm (4LD) and 150 μm (12LD) in a (a) linear and (b) nonlinear dispersive medium. (Inset) Comparison of the time-transformation (dashed red) and the FDTD (solid yellow) methods in the nonlinear case; an expanded view of the daughter pulse is also shown.

Equations (7)

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

Eout(t)=h(ttTd)Ein(t)dt,
Eout(t)=h(tt1)E(t1)J(t1)dt1,
Tnl(t)=n2LctR(tt)E2(t)dt,
Ein(t)=E0exp[t2/(2T02)]cos(2πfct).
Δn=12n2E02exp(t2/T02)[1+cos(4πfct)].
R(t)=τk1exp(t/τk).
ε(ω)=ε+ω02(εsε)ω02iδωω2.

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