Abstract

Self-focusing of intense laser beams and pulses of light in real nonlinear media is in general accompanied by material losses that require corrections to the conservative Nonlinear Schrödinger equations describing their propagation. Here we examine loss mechanisms that exist even in lossless media and are caused by shedding of energy away from the self-trapping beam making it to relax to an exact solution of lower energy. Using the conservative NLS equations with absorbing boundary conditions we show that energy shedding not only occurs during the initial reshaping process but also during oscillatory propagation induced by saturation of the nonlinear effect. For pulsed input we also show that, depending on the sign and magnitude of dispersion, pulse splitting, energy shedding, collapse or stable self-focusing may result.

© 2013 OSA

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References

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  1. G.A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP15, 1088–1090 (1962).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
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2013 (1)

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

2007 (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports441, 47–189 (2007).
[CrossRef]

2002 (1)

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

1999 (1)

1998 (2)

D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, “Internal modes of envelope solitons,” Physica D116, 121–142 (1998).
[CrossRef]

Y.S. Kivshar, “Bright and dark solitons in non-Kerr media,” Optical and Quantum Electronics30, 571–614 (1998).
[CrossRef]

1997 (1)

1996 (1)

J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
[CrossRef] [PubMed]

1993 (1)

H.S. Nalwa, “Organic materials for third-order nonlinear optics,” Advanced Materials5(5), 341–358 (1993).
[CrossRef]

1992 (1)

1991 (1)

1964 (1)

R. Chiao, E. Garmire, and C.H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett.13, 479–482 (1964).
[CrossRef]

1962 (1)

G.A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP15, 1088–1090 (1962).

Afanasjev, V.V.

D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, “Internal modes of envelope solitons,” Physica D116, 121–142 (1998).
[CrossRef]

Ahles, M.

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Askaryan, G.A.

G.A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP15, 1088–1090 (1962).

Boyd, R.W.

R.W. Boyd, Nonlinear Optics (Academic Press, 2003).

Chiao, R.

R. Chiao, E. Garmire, and C.H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett.13, 479–482 (1964).
[CrossRef]

Couairon, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports441, 47–189 (2007).
[CrossRef]

Denz, C.

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Durand, M.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Durcu, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Fibich, G.

Forget, N.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Gaeta, A.

J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
[CrossRef] [PubMed]

Gaeta, A.L.

Garmire, E.

R. Chiao, E. Garmire, and C.H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett.13, 479–482 (1964).
[CrossRef]

Gatz, S.

Grabielle, S.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Herrmann, J.

Houard, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Jarnac, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Kivshar, Y.S.

D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, “Internal modes of envelope solitons,” Physica D116, 121–142 (1998).
[CrossRef]

Y.S. Kivshar, “Bright and dark solitons in non-Kerr media,” Optical and Quantum Electronics30, 571–614 (1998).
[CrossRef]

Liu, Y.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

Mysyrowicz, A.

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports441, 47–189 (2007).
[CrossRef]

Nalwa, H.S.

H.S. Nalwa, “Organic materials for third-order nonlinear optics,” Advanced Materials5(5), 341–358 (1993).
[CrossRef]

Pelinovsky, D.E.

D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, “Internal modes of envelope solitons,” Physica D116, 121–142 (1998).
[CrossRef]

Petter, J.

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Ranka, J.

J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
[CrossRef] [PubMed]

Rothenberg, J.E.

Schirmer, R.

J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
[CrossRef] [PubMed]

Schröder, J.

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Sulem, C.

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 1999).

Sulem, P.-L.

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 1999).

Townes, C.H.

R. Chiao, E. Garmire, and C.H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett.13, 479–482 (1964).
[CrossRef]

Träger, D

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Weilnau, C.

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

Advanced Materials (1)

H.S. Nalwa, “Organic materials for third-order nonlinear optics,” Advanced Materials5(5), 341–358 (1993).
[CrossRef]

Ann. Phys. (Leipzig) (1)

C. Weilnau, M. Ahles, J. Petter, D Träger, J. Schröder, and C. Denz, “Spatial optical (2+1)-dimensional scalar- and vector-solitons in saturable nonlinear media,” Ann. Phys. (Leipzig)11(8), 573–629 (2002).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Optical and Quantum Electronics (1)

Y.S. Kivshar, “Bright and dark solitons in non-Kerr media,” Optical and Quantum Electronics30, 571–614 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

J. Ranka, R. Schirmer, and A. Gaeta, “Observation of Pulse Splitting in Nonlinear Dispersive Media,” Phys. Rev. Lett.77(18), 3783–3786 (1996).
[CrossRef] [PubMed]

M. Durand, A. Jarnac, A. Houard, Y. Liu, S. Grabielle, N. Forget, A. Durcu, A. Couairon, and A. Mysyrowicz, “Self-Guided Propagation of Ultrashort Laser Pulses in the Anomalous Dispersion Region of Transparent Solids: A New Regime of Filamentation,” Phys. Rev. Lett.110, 115003 (2013).
[CrossRef]

R. Chiao, E. Garmire, and C.H. Townes, “Self-Trapping of Optical Beams,” Phys. Rev. Lett.13, 479–482 (1964).
[CrossRef]

Physica D (1)

D.E. Pelinovsky, Y.S. Kivshar, and V.V. Afanasjev, “Internal modes of envelope solitons,” Physica D116, 121–142 (1998).
[CrossRef]

Physics Reports (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports441, 47–189 (2007).
[CrossRef]

Sov. Phys. JETP (1)

G.A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP15, 1088–1090 (1962).

Other (2)

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer, 1999).

R.W. Boyd, Nonlinear Optics (Academic Press, 2003).

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Figures (14)

Fig. 1
Fig. 1

(a) Reshaping of the self trapped beam from initially Gaussian input to a sech2 distribution. Energy is shed and oscillation in the amplitude is caused by the reshaping process, γ = 5.8. (b) Logarithmic intensity scale of (a), revealing the energy being shed upon entry to the medium.

Fig. 2
Fig. 2

Final power (blue) and resulting beam width (red) dependence on the nonlinear coefficient. From each self-trapped state, power loss during reshaping increases and the resulting beam width decreases.

Fig. 3
Fig. 3

Intensity (green) and normalized power (blue) for saturating NLS propagating 50 ZR. γ = 16, σ = 0.1. Here it can be seen that after the initial reshaping process there is a long lived damped oscillation in the amplitude rather than the stable soliton solution of the cubic NLS, and loss of power during propagation that is proportional to the oscillation in the width and amplitude.

Fig. 4
Fig. 4

Oscillations in the beam width and amplitude after 500 ZR of propagation in a saturating medium. γ = 200, σ = 10. (b) Logarithmic intensity scale of (a), showing shedding of low levels of energy during each defocusing cycle of the damped oscillations in saturable media.

Fig. 5
Fig. 5

Symmetry breaking instability of saturating NLS solution falling between solution branches. γ = 30, σ = 0.1.

Fig. 6
Fig. 6

Final power (top) and resulting beam width (bottom) dependence on the nonlinear coefficient for different values of σ. It can be seen that the point of self-trapping and hence the required critical power depends on the saturation parameter, requiring greater levels of power when there is greater saturation. At the point of exciting the second self-trapped state the solution becomes unstable and symmetry breaking instabilities occur.

Fig. 7
Fig. 7

(a) Oscillations in a 2D cw beam propagating in a saturable medium. r is the transverse radial variable. Intensities reached are roughly 6 times that of the 1D case. γ = 20, σ = 0.1. (b) Logarithmic intensity scale of (a), showing shedding of low levels of energy during defocusing, reducing the power of the beam.

Fig. 8
Fig. 8

Final power (blue) and resulting beam width (red) dependence on the nonlinear coefficient for a 2D beam in saturable media (σ = 0.1). As in the 1D case, increasing the nonlinear coefficient causes first self trapping (γ = 12.4) and then self-focusing. Unlike the 1D case the resulting beam width is significantly smaller, reducing to as much a 10% of the input, and the power lost due to energy shedding is increased, varying from 4% at the point of self-trapping up to 40%. Varying the saturation parameter again changes the length scales on which the same behavior occurs.

Fig. 9
Fig. 9

Left - evolution of the pulse profile on the temporal axis. Right - evolution of the profile on the spatial axis on entering a nonlinear medium. Without GVD there is no focusing or defocusing on the temporal axis, the splitting and reforming of the peak is due to oscillation along the spatial axis. Spatial evolution is similar to that of an equivalent 1D spatial simulation.

Fig. 10
Fig. 10

Left - initial pulse shape. Right - pulse shape after propagation to a stable profile. The front and back of the pulse have diffracted away, while the transverse slices with enough power to self-focus have become trapped.

Fig. 11
Fig. 11

Left - initial (red dash) and final (blue) temporal cross section. Right - initial and final spatial cross section. The pulse has narrowed in both dimensions during propagation, due to self-focusing in the spatial dimension, resulting in the sech2 distribution, but due to untrapped energy being shed in the temporal dimension, resulting in a parabolic profile.

Fig. 12
Fig. 12

Temporal (left) and spatial (right) evolution of a pulse in the normal GVD regime. β = +0.01, γ = 15. Due to the spreading of the pulse along the temporal dimension, the power in the spatial slices is reduced to the point at which no self-focusing can occur, resulting in collapse of the pulse.

Fig. 13
Fig. 13

(a) Evolution of the T/X profile in the anomalous-GVD regime. β = −1, γ = 20, Saturating nonlinearity. (b) Logarithmic intensity scale of (a). The results are similar to 2D spatial propagation of the pulse with a spatial dimension replaced with the temporal one

Fig. 14
Fig. 14

Final pulse shape after propagation to a stable profile, showing signiificantly reduced length and width. (a) β = −1, γ = 20, (b) β = −1, γ = 50

Equations (4)

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E ζ = i 2 2 E η 2 + i γ | E | 2 E
P = | E 0 | 2 d x = 1
E ζ = i 2 ( 2 E η 2 + 2 E ξ 2 ) + i γ | E | 2 1 + σ | E | 2 E
E ζ = i 2 2 E η 2 + i β 2 2 E T 2 + i γ | E | 2 1 + σ | E | 2 E

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