Abstract

The spatio-temporal dynamics of pulse propagation in dispersive Kerr media is studied based on the variational principle. By performing a Ritz optimization procedure with spatio-temporal Gaussian test functions, the equations of motion for the pulse parameters are derived. Comparison of these analytical results with numerically obtained solutions of the exact problem show very good agreement over a broad range of dispersion, diffraction, and nonlinearity parameters.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172–174 (1992).
    [CrossRef] [PubMed]
  2. M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
    [CrossRef]
  3. I. P. Christov, H. C. Kapteyn, M. M. Murnane, C.-P. Huang, and J. Zhou, “Space-time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
    [CrossRef] [PubMed]
  4. I. P. Christov and V. D. Stoev, “Kerr-lens mode-locked laser model: role of space-time effects,” J. Opt. Soc. Am. B 15, 1960–1966 (1998).
    [CrossRef]
  5. D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
    [CrossRef]
  6. D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
    [CrossRef]
  7. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [CrossRef]
  8. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
    [CrossRef]
  9. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [CrossRef]
  10. Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999).
    [CrossRef]

1999 (1)

1998 (2)

1995 (1)

1992 (1)

1991 (1)

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

1979 (2)

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

1975 (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Anderson, D.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Bonnedal, M.

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

Chen, M.

Chen, W.

Chen, Y.

Chernev, P.

Cho, S. H.

Christov, I. P.

Desaix, M.

Fujimoto, J. G.

Haus, H. A.

Huang, C.-P.

Huang, M.

Huang, W.

Ippen, E. P.

Kapteyn, H. C.

Kärtner, F. X.

Lisak, M.

M. Desaix, D. Anderson, and M. Lisak, “Variational approach to collapse of optical pulses,” J. Opt. Soc. Am. B 8, 2082–2086 (1991).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

Morgner, U.

Murnane, M. M.

Petrov, V.

Stoev, V. D.

Yu, L.

Zhou, J.

Zhu, Z.

J. Opt. Soc. Am. B (3)

Opt. Lett. (3)

Phys. Fluids (2)

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Prog. Quantum Electron. (1)

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[CrossRef]

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 (3)

Fig. 1
Fig. 1

Numerical results for no diffraction (F=0). Top: Pulse duration τ˜ (solid curves) and error εt (dotted curves). Bottom: Beam radius ρ˜ (solid curves) and error εr (dotted curves).

Fig. 2
Fig. 2

Numerical results for moderate diffraction (F=-2). Top: Pulse duration τ˜ (solid curves) and error εt (dotted curves). Bottom: Beam radius ρ˜ (solid curves) and error εr (dotted curves).

Fig. 3
Fig. 3

Numerical results for strong diffraction (F=-5). Top: Pulse duration τ˜ (solid curves) and error εt (dotted curves). Bottom: Beam radius ρ˜ (solid curves) and error εr (dashed curves).

Equations (62)

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

E(z, t, x, y)=2Z0n0 Re{U(z, t, x, y)×exp[j(ω0t-kz)]},
n=n0+n2I|U|2,
zU-jD 2tr2U+jB2x2+2y2U+jδ|U|2U=0,
s=z/L,
q=tr/T0,
g=x/x0,
h=y/y0,
---|u|2dgdhdqs=0=π3/2.
---|U|2dxdydtrs=0=E0,
u=π3/2T0x0y0E0 U.
S=L2T02 2k(ω)ω2ω=ω0=DLT02.
Fx=-L2kx02=-BLx02,
Fy=-L2ky02=-BLy02,
Φ=Lk0n2IE0π3/2T0x0y0=LδE0π3/2T0x0y0.
su-jS 2q2u-jFx 2g2u-jFy 2h2u+jΦ|u|2u=0.
δL=0,
L=Lu,u*, us, u*s, uq, u*q, ug, u*g, uh, u*hdsdqdgdh.
Lu*-dds Lu*s-ddq Lu*q-ddg Lu*g
-ddh Lu*h=0.
L=j2 u u*s-u* us+Suq2+Fxug2+Fyuh2+Φ2|u|4,
u(s, q, g, h)=A(s)exp-12τ(s)2+jb(s)q2+12ρx(s)2+jax(s)g2+12ρy(s)2+jay(s)h2,
--- |u|2dqdgdh=π3/2|A(s)|2τ(s)ρx(s)ρy(s).
τ-4S2τ3-SΦ2 1τ2ρxρy=0,
ρx-4Fx2ρx3-FxΦ2 1τρx2ρy=0,
ρy-4Fy2ρy3-FyΦ2 1τρxρy2=0.
τ(0)=ρx(0)=ρy(0)=1,
τ(0)=τ0,ρx(0)=ρx0,ρy(0)=ρy0.
b=-14S ττ,
ax=-14Fx ρxρx,
ay=-14Fy ρyρy.
|A(s)|=1τ(s)ρx(s)ρy(s).
dϕ(s)ds=-Sτ(s)2-Fxρx(s)2-Fyρy(s)2-782Φ|A(s)|2.
ρ+Kρ3=0,
ρ(s)=1-K×s2.
ssf,Ritz=K-1/2,
Φ(-F)=42.
E0=22π3/2T0n0k02n2I.
L=12[jAA*-jA*A-2|A|2(bq2+axg2+ayh2)]×exp-q2τ2+g2ρx2+h2ρy2+4|A|2Sq214τ4+b2+Fxg214ρx4+ax2+Fyh214ρy4+ay2exp-q2τ2+g2ρx2+h2ρy2+Φ2|A|4 exp-2q2τ2+g2ρx2+h2ρy2.
δL(A, A*, ax, ay, b, τ, ρx, ρy, A,
A*, ax, ay, b)ds=0,
L=---Ldqdgdh=π3/2j2(AA*-A*A)τρxρy-12|A|2(bτ3ρxρy+axτρx3ρy+ayτρxρy3)+12|A|2Sρxρy1τ+4b2τ3+Fxτρy1ρx+4ax2ρx3+Fyτρx1ρy+4ay2ρy3+Φ |A|442τρxρy.
LA-dds LA=0
j2 dds(A*τρxρy)+j2A*τρxρy-12A*(bτ3ρxρy+axτρx3ρy+ayτρxρy3)+12A*Sρxρy1τ+4b2τ3+Fxτρy1ρx+4ax2ρx3+Fyτρx1ρy+4ay2ρy3+Φ22|A|2A*τρxρy=0,
Lτ-dds Lτ=0
j2(AA*-A*A)+12|A|2-3bτ2-axρx2-ayρy2+S-1τ2+12b2τ2+Fx1ρx2+4ax2ρx2+Fy1ρy2+4ay2ρy2+Φ42|A|4=0,
Lρx-dds Lρx=0
j2(AA*-A*A)+12|A|2-bτ2-3axρx2-ayρy2+S1τ2+4b2τ2+Fx-1ρx2+12ax2ρx2+Fy1ρy2+4ay2ρy2+Φ42|A|4=0,
Lρy-dds Lρy=0
j2(AA*-A*A)+12|A|2-bτ2-axρx2-3ayρy2+S1τ2+4b2τ2+Fx1ρx2+4ax2ρx2+Fy-1ρy2+12ay2ρy2+Φ42|A|4=0,
Lb-dds Lb=0
12 dds(|A|2τ3ρxρy)+4S|A|2bτ3ρxρy=0,
Lax-dds Lax=0
12 dds(|A|2τρx3ρy)+4Fx|A|2axτρx3ρy=0,
Lay-dds Lay=0
12 dds(|A|2τρxρy3)+4Fy|A|2ayτρxρy3=0.
dds(|A|2τρxρy)=0
|A(s)|2τ(s)ρx(s)ρy(s)=|A(0)|2τ(0)ρx(0)ρy(0)=1.
j2(AA*-A*A)+12|A|2
×-bτ2-axρx2-ayρy2+S1τ2+4b2τ2+Fx1ρx2+4ax2ρx2+Fy1ρy2+4ay2ρy2+Φ2|A|2=0.
b=-14S ττ.
b+S1τ4-4b2+Φ42 |A|2τ2=0,
τ-4S2τ3-SΦ2 1τ2ρxρy=0.

Metrics