Abstract

We present a time-transformation approach for studying the propagation of optical pulses inside a nonlinear medium. Unlike the conventional way of solving for the slowly varying amplitude of an optical pulse, our new approach maps directly the input electric field to the output one, without making the slowly varying envelope approximation. Conceptually, the time-transformation approach shows that the effect of propagation through a nonlinear medium is to change the relative spacing and duration of various temporal slices of the pulse. These temporal changes manifest as self-phase modulation in the spectral domain and self-steepening in the temporal domain. Our approach agrees with the generalized nonlinear Schrödinger equation for 100 fs pulses and the finite-difference time-domain solution of Maxwell’s equations for two-cycle pulses, while producing results 20 and 50 times faster, respectively.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
    [CrossRef]
  2. F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  4. T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
    [CrossRef]
  5. Y. Xiao, G. P. Agrawal, and D. N. Maywar, Opt. Lett. 36, 505 (2011).
    [CrossRef]
  6. M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R) (2006).
    [CrossRef]
  7. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
    [CrossRef]
  8. C. V. Hile and W. L. Kath, J. Opt. Soc. Am. B 13, 1135 (1996).
    [CrossRef]

2011 (1)

2006 (1)

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R) (2006).
[CrossRef]

1997 (1)

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

1996 (1)

1995 (1)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

1967 (2)

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
[CrossRef]

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Agrawal, G. P.

Y. Xiao, G. P. Agrawal, and D. N. Maywar, Opt. Lett. 36, 505 (2011).
[CrossRef]

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

Brabec, T.

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

DeMartini, F.

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Gustafson, T. K.

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Hile, C. V.

Kath, W. L.

Kelley, P. L.

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Kivshar, Y. S.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Krausz, F.

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

Maywar, D. N.

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Mitsugi, S.

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R) (2006).
[CrossRef]

Notomi, M.

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R) (2006).
[CrossRef]

Shimizu, F.

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
[CrossRef]

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Townes, C. H.

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Xiao, Y.

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

Mod. Phys. Lett. B (1)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, Phys. Rev. 164, 312 (1967).
[CrossRef]

Phys. Rev. A (1)

M. Notomi and S. Mitsugi, Phys. Rev. A 73, 051803(R) (2006).
[CrossRef]

Phys. Rev. Lett. (2)

F. Shimizu, Phys. Rev. Lett. 19, 1097 (1967).
[CrossRef]

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

Other (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

(a) Time-transformation performed by a medium for Gaussian pulse propagation and (b) the corresponding scaling factor s. Cases: n2>0 (solid); n2<0 (dashed); n2=0 (dotted).

Fig. 2.
Fig. 2.

Output shape and spectrum of 10 ps (left column) and 100 fs (right column) Gaussian pulse obtained with our approach (solid lines) and using the NLS equation (dotted lines). Dashed curve shows input spectra in each case.

Fig. 3.
Fig. 3.

Schematic propagation of the electric field of a 2-cycle pulse through a Kerr medium (n2>0). Dotted curves show pulse envelope. Red time slices at the bottom move toward the back of the pulse.

Fig. 4.
Fig. 4.

Nonlinear propagation of a 2-cycle Gaussian pulse (solid: time-transformation; dotted: FDTD). Dashed curve shows the input electric field.

Equations (7)

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

Eout(t)=δ[ttTr(t)]Ein(t)dt,
t=t+Tr(t),
Tr(t)=Tr0+n2I0f(t)L/c,
Eout(t)=δ[ttTr0τnf(t)]Ein(t)dt.
Eout(t)=[dt/dt]Ein(t)=s(t)Ein(t),
Az=iγ|A|2A,
Az=iγ|A|2Aγω0|A|2At.

Metrics