## Optical multi-hysteresises and quasi-solitons in nonlinear plasma |

Optics Express, Vol. 21, Issue 11, pp. 13134-13144 (2013)

http://dx.doi.org/10.1364/OE.21.013134

Acrobat PDF (879 KB)

### Abstract

An overdense plasma layer irradiated by intense light can exhibit dramatic nonlinear-optical effects due to a relativistic mass-effect of free electrons: highly-multiple hysteresises of reflection and transition, and emergence of immobile waves of large amplitude. Those are trapped quasi-soliton spikes sustained by a weak pumping having a tiny fraction of their peak intensity once they have been excited first by higher power pumping. The phenomenon persists even in the layers with ”soft”, wash-out boundaries, as well as in a semi-infinite plasma with low absorption.

© 2013 OSA

## 1. Introduction

*nonlinear*optics: a nonlinear refractive index can cause a phase-transition-like effect, since a small light-induced change may translate into a switch from full reflection to full transmission, resulting in a huge hysteresis. Predicted in [1, 2] for nonlinear interfaces, it was explored experimentally in [3

3. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. **35**, 846–848 (1979) [CrossRef] .

6. S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B **49**, 441–444 (1989) [CrossRef] .

3. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. **35**, 846–848 (1979) [CrossRef] .

4. P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE **17**, 340–348 (1981) [CrossRef] .

6. S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B **49**, 441–444 (1989) [CrossRef] .

5. P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE **20**, 30–36 (1984) [CrossRef] .

9. J. H. Marburger and R. F. Tooper, “Nonlinear optical standing waves in overdense plasmas, ” Phys. Rev. Lett ., **35**, 1001–1004 (1975) [CrossRef] .

10. S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B **4**, 1935–1942 (1992) [CrossRef] .

15. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. **78**, 309–371 (2006), and references therein [CrossRef] .

16. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. **58**, 160–163 (1987) [CrossRef] [PubMed] .

20. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. **90**, 053902 (2003) [CrossRef] [PubMed] .

21. O. Zobay, S. Potting, P. Meystre, and E. M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A **59**, 643–648 (1999) [CrossRef] .

*standing, immobile*quasi-solitons instead of propagating ones) due to self-induced retro-reflection. The new property is that for the

*same incident power*an EM-wave can penetrate into a nonlinear material to

*different depths*– varying by orders of magnitude – depending of the history of pumping. We assume a stationary,

*cw*, or long pulse mode, and use only RL-nonlinearity in a cold plasma. While this model is greatly simplified

*vs*various kinetic approaches, it allows us to keep the basic features necessary to elucidate new results, and have the theory applicable to other systems. Remarkably, even a few-

*λ*-thick plasma layer can produce the effect, so that the absorption and diffraction of light (including nonlinear self-focusing) would be unlikely to significantly affect the propagation. Indeed, the lowest thickness of the layer needed to observe the effect is of the same order as the size of single soliton formed in the layer for a given a normalized detuning from critical frequency

*δ*, which is

*δ*= 10

^{−4}, one has that size amounting to ≈ 0.225

*mm*at

*λ*≈ 10

*μ*

*m*(

*CO*

_{2}laser), which is far below characteristic absorption or self-focusing scales.

22. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., **36**, 613–614 (1980) [CrossRef] .

24. D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. **64**, 1721–1724 (1990) [CrossRef] [PubMed]

22. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., **36**, 613–614 (1980) [CrossRef] .

23. A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quant. Electr. , **QE-21**: 1529–1537 (1985) [CrossRef] .

24. D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. **64**, 1721–1724 (1990) [CrossRef] [PubMed]

25. A. E. Kaplan and Y. J. Ding, “Hysteretic and multiphoton optical resonances of a single cyclotron electron,” IEEE JQE **24**, 1470–1482 (1988), and references therein [CrossRef] .

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , **105**, 675–686 (2007) [CrossRef]

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , **105**, 675–686 (2007) [CrossRef]

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , **105**, 675–686 (2007) [CrossRef]

**105**, 675–686 (2007) [CrossRef]

*semi-infinite*plasma with a small absorption, which also develops a strong retro-reflection. The ”Sommerfeld condition” (no wave comes from the ”infinity”) is to be revisited here: a wave

*is back-reflected*deep inside the plasma and comes to the boundary. Interfering with a forward wave, it results in a semi-standing wave, trapped QS’s, and multistability, same as in a finite layer. The energy accumulated in QS’s and excited free electrons, can then be released if plasma density reduces, as it may happen, e. g. in an astrophysical environment.

## 2. Nonlinear relativistic wave equation and boundary conditions

*z*-axis, we have ∇

^{2}=

*d*

^{2}/

*dz*

^{2}. For semi-infinite dielectrics, a EM-wave incident from a dielectric with

*ε*=

*ε*> 0 under the angle

_{in}*θ*onto a material of

*ε*=

*ε*> 0,

_{NL}*ε*in Eq. (2) is replaced by

*ε*[

_{in}*ε*(

_{NL}*ω*)/

*ε*(

_{in}*ω*) − sin

^{2}

*θ*]. For a microwave (

*mw*) waveguide with a critical frequency

*ω*,

_{wg}*ε*in Eq. (2) is replaced by

*ε*= 0. Equation (2) reduces then to where

*ζ*=

*kz*, and ”prime” denotes

*d/d*

*ζ*; in general, we do not assume

*ε uniform*in

*ζ*-axis. In a weakly-nonlinear media one can break the field into counter-propagating traveling waves and find their amplitudes

*via*boundary conditions. However, near a crossover point one in general cannot distinguish between those waves. To make no assumptions whether a wave is traveling, standing, or mixed, we represent the field using real variables

*u*, and phase (eikonal),

*ϕ*, as Since

*ℰ*is in general complex, while

*ε*=

*ε*(

*u*

^{2}), Eq. (4), is isomorphous to a 3-rd order equation for

*u*; yet, it is fully integrable in quadratures. Its first integral is a scaled momentum flux In a lossless media

*P*is conserved over the entire space

*ζ*< ∞, even if the medium is nonuniform, multi-layered, linear and/or nonlinear, etc. If a layer borders a dielectric of

*ε*=

*ε*at the exit, we have

_{ex}*u*: which makes an unusual yet greatly useful tool. Since it deals only with a real amplitude and uses flux

*P*as a parameter, Eq. (7) is nonlinear even for a

*linear propagation*, yet is still analytically solvable if a density

*ρ*is uniform across the layer (

_{e}*∂ε/∂*

*ζ*= 0). A full-energy-like invariant of Eq. (7) is where

*u*′

^{2}/2 is ”kinetic”, and

*U*– ”potential” energies. For a RL-nonlinearity, Eq. (3), we have Here

*W*is a scaled free EM energy density of

*ε*-nonlinear medium [35]

*H*is magnetic field. If a layer exit wall is a dielectric, one has

*u*′

*= 0 (see below). For a metallic mirror,*

_{ex}*u*= 0; and

_{ex}*W*= 0 for an evanescent wave in a semi-infinite medium. The implicit solution for spatial dynamics of

*u*in general case is found now as Boundary conditions at the borders with linear dielectrics at the entrance,

*ζ*= 0, with

*ε*=

*ε*, and at the exit,

_{in}*ζ*=

*d*, with

*ε*=

*ε*, result in complex amplitudes of incident,

_{ex}*ℰ*, and reflected,

_{in}*ℰ*, waves at

_{rfl}*ζ*= 0: where ”+” corresponds to

*ℰ*, and ”−” – to

_{in}*ℰ*. At the exit point,

_{rfl}*ζ*=

*d*, we have

*u*=

*ℰ*

_{ex}*= u*;

_{ex}*u*′ = 0; and

*P*= 0 corresponds to full reflection, resulting in either strictly standing wave, or nonlinear evanescent wave in a semi-infinite plasma, in particular in a ”standing” soliton-like solution (see below). If

*ε*(

*u*

^{2}= 0) < 0, there are no

*linear*traveling waves. Yet a purely traveling

*nonlinear*wave may exist at sufficiently strong intensity

*ε*(

*u*= 0) < 0, it is strongly unstable. A non-periodic solution of Eq. (7) with

*P*= 0 is a nonlinear evanescent wave that forms a standing, trapped soliton. In low-RL case, one needs a small detuning from crossover point,

*δ*≡ 1 −

*ν*≪ 1, to attain the effect at low laser intensity,

*u*

^{2}≪ 1, so that the the dielectric constant, Eq. (4), is Kerr-like and small:

*ε*≈ −2

_{rl}*δ*+

*u*

^{2}/2, |

*ε*

_{rl}*|*≪ 1. A full solution of Eq. (7) with

*P*= 0 and

*u*→ 0 at

*ζ*→ ∞ yields then a standing soliton with a familiar intensity profile: where the peak location

*ζ*is an integration constant. For an arbitrary frequency,

_{pk}*ν*< 1, the soliton peak intensity is instead of 8

*δ*as in Eq. (13); and When

*ν*

^{2}< 1/2, it is a strongly-RL soliton,

## 3. Finite layer plasma and quasi-solitons

*ε*≈ −2

_{rl}*δ*+

*u*

^{2}/2 yields then elliptic integrals of imaginary argument of the first kind; more importantly, Eq. (7), and its invariant avails themselves to detailed analysis. The numerical simulations are needed, however, to find a solution for (a) strongly-RL field [using Eq. (7) and its integrals], or (b) non-uniform plasma density in Eq. (7) (Section 4 below), or(c) plasma with absorption [Eq. (4) with

*ν*

^{2}replaced with

*ν*

^{2}(1 +

*i*

*α*), where

*α*is an absorption factor, Section 6 below). It is then found by an ”inverse propagation” procedure, whereby we essentially back-track the propagation from a purely traveling exit wave back to the entrance. One sets first a certain magnitude of

*u*′

*= 0 at the exit, numerically computes an amplitude profile*

_{ex}*u*(

*ζ*) back to the entrance and incident and reflected intensities

*vs*incident intensity,

*P*is found then with a single run,

*vs*a so called multi-shooting commonly used in search of solution with conditions set at two boundaries. This provides a very fast numerical simulation

*vs*multi-shooting; besides, the latter one is very unreliable when dealing with apriory unknown number of multi-solutions.

*L*= 10

*λ*, where

*L*is the layer thickness, Fig. 1 show the emergence of large number,

*N*, of huge hysteretic loops of the transmission (same as in reflection, not shown here), which bounces between full transparency (near the points touching an

_{hs}*FT*line) to nearly full reflection (near the points touching an envelope

*NFR*). In general,

*N*=

_{hs}*O*(

*L/*

*λ*). In an unbound plasma, the solution of Eq. (7) with a traveling component,

*P*> 0, is a spatially periodic and positively defined, with the intensity,

*u*

^{2}(

*ζ*), bouncing between two limits,

*P*/16 ≪

*δ*

^{2}≪ 1, we have i. e. the peaks are relatively large,

*P*and

*P*> 1, we have as for a standing, albeit inhibited wave component in free space, while traveling wave component emerges dominant, resulting in self-induced transparency.

## 4. Relativistic multi-hysteresises and immobile quasi-solitons

*ζ*= 0. The valley,

*ζ*= 0, as

*ζ*≈ 0. With further slight increase of pumping, it gets unsustainable, and the field configuration has to jump up to the next stable branch of excitation, whereby it forms a steady QS at the back of the layer. If after that

_{pk}*Q*, – can be tremendously high: where

*N*is the number of QS’s in a layer; the one with

*N*= 1 occurs after the first jump-up. In the example for Fig. 1 (

*δ*= 0.02,

*L*= 10

*λ*),

*Q*∼ 10

^{7}. In semi-infinite plasma,

*Q*is limited by absorption, see below. It also decreases as

*N*increases; the field profile for

*N*= 2, is depicted in Fig. 1, point 3, curve 3 in the inset. Only half of multi-steady-states are stable; the stability condition is that the EM-energy density increases with the pumping, i. e.

*N*-th stable branch as a

*N*-th mode of a self-induced resonator, with full transparency points marking the resonance. The plot of transmission intensity,

*vs*incident intensity,

**FT**line, whereby

**NTR**). The envelope of the latter one, i. e. the minimal transmitted intensity,

## 5. No need for sharp boundaries!

36. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabri-Perot interferometer,” Phys. Rev. Lett. **36**, 1135–1138 (1976) [CrossRef] .

**105**, 675–686 (2007) [CrossRef]

*ρ*, varies smoothly in space, and vanishes completely at far edges of a plasma layer. Our numerical simulations using Eqs. (4) and (7), where we have now to make

*ε*explicitly dependent on the distance

*ζ*: showed that a layer with ”soft” shoulders making ∼ 50% of the entire layer length, still exhibits a few hysteresises, and a large number of self-induced resonances. An example of this for

*δ*= 10

^{−4}is shown at Fig. 2, depicting transmitted and reflected wave intensities

*vs*the incident wave intensity in the case whereby the plasma layer has a varying plasma density along the propagation path, which has a flat density distribution in the middle and tapers down to zero at the edges of the layer, see inset at Fig. 2. For numerical simulation purposes, a specific model profile chosen by us here is

*ρ*(

*z*)/

*ρ*

_{0}= 1 at

*|z|*≤

*L*/2, and

*ρ*(

*z*) = 0 elsewhere, where

*L*is the thickness of a sub-layer with a flat distribution of plasma density, and

*S*is the shoulder thickness; at the half-max of the density distribution the layer is

*L*+

*S*thick. In the example on Fig. 2,

*L*=

*S*= 10

*λ*. One can see that even at such smoothly distributed plasma there is still two hysteresises (the largest one of them is indicated by vertical arrows) and multiple self-induced resonances. The formation of QS field spikes at the slopes density distribution here might be assisted by a linear so called ”gradient marker” effect [38

38. A. E. Kaplan, “Gradient marker” – a universal wave pattern in inhomogeneous continuum”, Phys. Rev. Lett., **109**, 153901(1–5) (2012) [CrossRef]

## 6. Multi-hysteresises in semi-infinite plasma with absorption

*semi-infinite*plasma. Only two kind of waves [1, 2] in a lossless case satisfy then the Sommerfeld condition – no wave ”comes back” from

*ζ*→ ∞ – a traveling, Eq. (12),

*du/d*

*ζ*→ 0, and an evanescent, Eq. (13), waves,

*u*→ 0. Our preliminary investigation of Eq. (2) showed that the wave, Eq. (12), is unstable both in 2D&3D-propagation – and, of Eq. (1) – in temporal domain in 1D-case. However, using Eq. (4), in which a real term

*ν*

^{2}is replaced by a complex one: where

*α*is an absorption factor, and

*τ*is an electron momentum relaxation time, one can show that even a

*steady*1D-wave, Eq. (12), does not survive small absorption,

*α*≪ 1. A condition for a hysteresis to emerge is then where

*α*is a critical absorption [37]. Near

_{cr}*α*∼

*α*, a jump-up occurs at

_{cr}*Q*≈

*δ*

*/*

*α*, and can still be huge. If

*α*≪

*α*, the first jump still occurs at the incident intensity equal to a quarter of soliton, Eq. (13), intensity: When pumped hard, new QS emerge and move deeper into plasma. A question then is what is the spacing

_{cr}*L*, between the most submerged ones, i. e. the maximum spacing between standing solitons in absorbing plasma. We found that the ratio ”

_{spc}*min/max*” of the intensities is Assuming that down to the bottom, the intensity follows very close to the soliton solution, Eq. (13), and making correction for the length of the ”bottom”, we estimate the spacing

*L*as which was confirmed by our numerical simulations. The QS’s are well spaced and distinguished from each other if

_{spc}*α*≪

*δ*, which is a natural condition for the detuning

*δ*to be set sufficiently far from the the cross-over point in the presence of non-zero absorption.

## 7. Discussion

*CO*

_{2}laser, with a gas density controlled to reach a crossover point at

*λ*

_{CO2}≈ 10

*μ*

*m*. This process may be also naturally occurring in astrophysical environment in plasma sheets expelled from a star (e. g. the Sun); part of the star’s radiation spectrum below the initial plasma frequency is powerful enough to penetrate into the layer and be trapped as QS’s. When the layer expands, they get released as a burst of radiation, similarly to bubbles in boiling water. It is also conceivable that the QS trapping and consequent release may be part of the physics of ball-lighting subjected to a powerful radiation emitted by the main lighting discharge in

*mw*and far infrared domains. The QS’s might be used e. g. for laser fusion to deposit laser power much deeper into the fusion pallets; or for heating the ionosphere layers by a powerful

*rf*radiation.

## 8. Conclusions

## 9. Acknowledgments

## References and links

1. | A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. |

2. | A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP |

3. | P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. |

4. | P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE |

5. | P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE |

6. | S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B |

7. | A. I. Akhiezer and R. V. Polovin, “Theory of wave motion of an electron plasma” Sov. Phys. JETP-USSR , |

8. | C. Max and F. Perkins, “Strong electromagnetic waves in overdense plasmas” Phys. Rev. Lett ., |

9. | J. H. Marburger and R. F. Tooper, “Nonlinear optical standing waves in overdense plasmas, ” Phys. Rev. Lett ., |

10. | S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B |

11. | T. Z. Esirkepov, F. F. Kamenets, S. V. Bulanov, and N. M. Naumova, “Low-frequency relativistic electromagnetic solitons in collisionless plasmas,” JETP Lett. |

12. | M. Tushentsov, A. Kim, F. Cattani, D. Anderson, and M. Lisak, “Electromagnetic energy penetration in the self-induced transparency regime of relativistic laser-plasma interactions,” Phys. Rev. Lett. |

13. | T. Esirkepov, K. Nishihara, S. V. Bulanov, and F. Pegoraro, “Three-dimensional relativistic electromagnetic subcycle solitons,” Phys. Rev. Lett. |

14. | G. Lehmann, E. W. Laedke, and K. H. Spatchek, “Two-dimensional dynamics of relativistic solitons in cold plasmas,” Phys. Plasma |

15. | G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. |

16. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. |

17. | J. E. Sipe and H. G. Winful, “Nonlinear Schroedinger solitons in periodic structure,” Opt. Lett. |

18. | D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. |

19. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

20. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. |

21. | O. Zobay, S. Potting, P. Meystre, and E. M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A |

22. | H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., |

23. | A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quant. Electr. , |

24. | D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. |

25. | A. E. Kaplan and Y. J. Ding, “Hysteretic and multiphoton optical resonances of a single cyclotron electron,” IEEE JQE |

26. | A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , |

27. | A. E. Kaplan, “Hysteresis in cyclotron resonance based on weak-relativistic mass-effect of the electron,” Phys. Rev. Lett. |

28. | G. Gabrielse, H. Dehmelt, and W. Kells, “Observation of a relativistic bistable hysteresis in the cyclotron motion of a single electron,” Phys. Rev. Lett. |

29. | A. E. Kaplan, “Relativistic nonlinear optics of a single cyclotron electron,” Phys. Rev. Lett. |

30. | A. E. Kaplan and P. L. Shkolnikov, “Lasetron: a proposed source of powerful nuclear-time-scale electromagnetic bursts,” Phys. Rev. Lett. |

31. | R. J. Noble, “Plasma wave generation in the beat-wave accelerator,” Phys. Rev. A |

32. | A. B. Shvartsburg, “Resonant Joule phenomena in a magnetoplasma”, Phys. Reports , |

33. | B. M. Ashkinadze and V. I. Yudson, “Hysteretic microwave cyclotron-like resonance in a laterally confined two-dimensional electron gas,” Phys. Rev. Lett. |

34. | G. Shvets, “Beat-wave excitation of plasma waves based on relativistic bistability,” Phys. Rev. Lett. |

35. | B.Ya. Zeldovich, “Nonlinear optical effects and the conservation laws,” Brief Comm. Physics, Lebedev Inst. (FIAN), Moscow |

36. | H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabri-Perot interferometer,” Phys. Rev. Lett. |

37. | This suggests an explanation of a hysteresis lacking in [5]: an artificial nonlinearity used there was due to small dielectric particles suspended in a liquid and had large dissipation because of strong light scattering, |

38. | A. E. Kaplan, “Gradient marker” – a universal wave pattern in inhomogeneous continuum”, Phys. Rev. Lett., |

39. | G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms” Sov. Phys., JETP-USSR , |

40. | V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids” JETP Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.1450) Nonlinear optics : Bistability

(350.5720) Other areas of optics : Relativity

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 26, 2013

Revised Manuscript: May 5, 2013

Manuscript Accepted: May 7, 2013

Published: May 21, 2013

**Citation**

A. E. Kaplan, "Optical multi-hysteresises and quasi-solitons in nonlinear plasma," Opt. Express **21**, 13134-13144 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13134

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### References

- A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett.24, 114–119 (1976).
- A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP45, 896–905 (1977).
- P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett.35, 846–848 (1979). [CrossRef]
- P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE17, 340–348 (1981). [CrossRef]
- P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE20, 30–36 (1984). [CrossRef]
- S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B49, 441–444 (1989). [CrossRef]
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